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SUMMARY TECHNICAL REPORT 
OF THE 

NATIONAL DEFENSE RESEARCH COMMITTEE 


Manuscript and illustrations for this volume were prepared for 
publication by the Summary Reports Group of the Columbia 
University Division of War Research under contract OEMsr-1131 
with the Office of Scientific Research and Development. This vol¬ 
ume was printed and bound by the Columbia University Press. 


Distribution of the Summary Technical Report of NDRC has 
been made by the War and Navy Departments. Inquiries concern¬ 
ing the availability and distribution of the Summary Technical 
Report volumes and microfilmed and other reference material 
should be addressed to the War Department Libra^q Room 
1A-522, The Pentagon, Washington 25, D. C., or to the Office of 
Naval Research, Navy Department, Attention: Reports and 
Documents Section, Washington 25, D. C. 


Copy No. 

258 


This volume, like the seventy others of the Summary Technical 
Report of NDRC, has been written, edited, and printed under 
great pressure. Inevitably there are errors which have slipped past 
Division readers and proofreaders. There may be errors of fact not 
known at time of printing. The author has not been able to follow 
through his w T riting to the final page proof. 


Please report errors to: 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATA) 
WASHINGTON 25, D. C. 


A master errata sheet will be compiled from these reports and sent 
to recipients of the volume. Your help will make this book more 
useful to other readers and will be of great value in preparing any 
revisions. 


SUMMARY TECHNICAL REPORT OF THE 
COMMITTEE ON PROPAGATION, NDRC 

VOLUME 3 


THE PROPAGATION 
OF RADIO WAVES THROUGH 
THE STANDARD ATMOSPHERE 


OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 
VANNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

COMMITTEE ON PROPAGATION 
CHAS. R. BURROWS, CHAIRMAN 


WASHINGTON, D.C., 1946 



NATIONAL DEFENSE RESEARCH COMMITTEE 


James B. Conant, Chairman 
Richard C. Tolman, Vice Chairman 
Roger Adams Army Representative 1 

Frank B. Jewett Navy Representative 2 

Karl T. Compton Commissioner of Patents 3 

Irvin Stewart, Executive Secretary 


1 Army representatives in order of service: 

Maj. Gen. G. V. Strong Col. L. A. Denson 

Maj. Gen. R. C. Moore Col. P. R. Faymonville 

Maj. Gen. C. C. Williams Brig. Gen. E. A. Regnier 

Brig. Gen. W. A. Wood, Jr. Col. M. M. Irvine 

Col. E. A. Routheau 


2 Navy representatives in order of service: 

Rear Adm. H. G. Bowen Rear Adm. J. A. Furer 

Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren 

Commodore H. A. Schade 
3 Commissioners of Patents in order of service: 
Conway P. Coe Casper W. Ooms 


NOTES ON THE ORGANIZATION OF NDRC 


The duties of the National Defense Research Committee 
were (1) to recommend to the Director of OSRD suitable 
projects and research programs on the instrumentalities of 
warfare, together with contract facilities for carrying out 
these projects and programs, and (2) to administer the 
technical and scientific work of the contracts. More specifi¬ 
cally, NDRC functioned by initiating research projects on 
requests from the Army or the Navy, or on requests from an 
allied government transmitted through the Liaison Office 
of OSRD, or on its own considered initiative as a result of 
the experience of its members. Proposals prepared by the 
Division, Panel, or Committee for research contracts for 
performance of the work involved in such projects were 
first reviewed by NDRC, and if approved, recommended to 
the Director of OSRD. Upon approval of a proposal by the 
Director, a contract permitting maximum flexibility of 
scientific effort was arranged. The business aspects of the 
contract, including such matters as materials, clearances, 
vouchers, patents, priorities, legal matters, and administra¬ 
tion of patent matters were handled by the Executive 
Secretary of OSRD. 

Originally NDRC administered its work through five 
divisions, each headed by one of the NDRC members. 
These were: 

Division A — Armor and Ordnance 
Division B — Bombs, Fuels, Gases & Chemical Problems 
Division C — Communication and Transportation 
Division D — Detection, Controls, and Instruments 
Division E — Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three 
administrative divisions, panels, or committees were cre¬ 
ated, each with a chief selected on the basis of his outstand¬ 
ing work in the particular field. The NDRC members then 
became a reviewing and advisory group to the Director of 
OSRD. The final organization was as follows: 

Division 1 — Ballistic Research 

Division 2 — Effects of Impact and Explosion 

Division 3 — Rocket Ordnance 

Division 4 — Ordnance Accessories 

Division 5 — New Missiles 

Division 6 — Sub-Surface Warfare 

Division 7 — Fire Control 

Division 8 — Explosives 

Division 9 — Chemistry 

Division 10 — Absorbents and Aerosols 

Division 11 — Chemical Engineering 

Division 12 — Transportation 

Division 13 — Electrical Communication 

Division 14 — Radar 

Division 15 — Radio Coordination 

Division 16 — Optics and Camouflage 

Division 17 — Physics 

Division 18 — War Metallurgy 

Division 19 — Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on Propagation 

Tropical Deterioration Administrative Committee 



NDRC FOREWORD 


As events of the years preceding 1940 revealed 
ii more and more clearly the seriousness of the 
world situation, many scientists in this country 
came to realize the need of organizing scientific 
research for service in a national emergency. Recom¬ 
mendations which they made to the White House 
were given careful and sympathetic attention, and 
as a result the National Defense Research Com¬ 
mittee [NDRC] was formed by Executive Order 
of the President in the summer of 1940. The members 
of NDRC, appointed by the President, were in¬ 
structed to supplement the work of the Army and 
the Navy in the development of the instrumentalities 
of war. A year later upon the establishment of the 
Office of Scientific Research and Development 
[OSRD], NDRC became one of its units. 

The Summary Technical Report of NDRC is a 
conscientious effort on the part of NDRC to sum¬ 
marize and evaluate its work and to present it in a 
useful and permanent form. It comprises some 
seventy volumes broken into groups corresponding 
to the NDRC Divisions, Panels, and Committees. 

The Summary Technical Report of each Division, 
Panel, or Committee is an integral survey of the 
work of that group. The first volume of each group’s 
report contains a summary of the report, stating 
the problems presented and the philosophy of 
attacking them, and summarizing the results of the 
research, development, and training activities under¬ 
taken. Some volumes may be “state of the art” 
treatises covering subjects to which various research 
groups have contributed information. Others may 
contain descriptions of devices developed in the 
laboratories. A master index of all these divisional, 
panel, and committee reports which together con¬ 
stitute the Summary Technical Report of NDRC is 
contained in a separate volume, which also includes 
the index of a microfilm record of pertinent technical 
laboratory reports and reference material. 

Some of the NDRC-sponsored researches which 
had been declassified by the end of 1945 were of 
sufficient popular interest that it was found desirable 
to report them in the form of monographs, such as 
the series on radar by Division 14 and the monograph 
on sampling inspection by the Applied Mathematics 


Panel. Since the material treated in them is not 
duplicated in the Summary Technical Report of 
NDRC, the monographs are an important part of 
the story of these aspects of NDRC research. 

In contrast to the information on radar, which is of 
widespread interest and much of which is released 
to the public, the research on subsurface warfare is 
largely classified and is of general interest to a more 
restricted group. As a consequence, the report of 
Division 6 is found almost entirely in its Summary 
Technical Report, which runs to over twenty vol¬ 
umes. The extent of the work of a division cannot 
therefore be judged solely by the number of volumes 
devoted to it in the Summary Technical Report of 
NDRC: account must be taken of the monographs 
and available reports published elsewhere. 

Though the Committee on Propagation had a 
comparatively short existence, being organized 
rather late in the war program, its accomplishments 
were definitely effective. That so many individuals 
and organizations worked together so harmoniously 
and contributed so willingly to the Committee’s 
efforts is a tribute to the leadership of the Chairman, 
Charles R. Burrows. The latest information in this 
field was gathered from the four corners of the earth, 
organized, and dispatched to the points where it 
would aid most in the prosecution of the war. 

Much credit must be given, not only to the mem¬ 
bers of the Committee and its contractors, but also 
to the many other individuals who gave so gener¬ 
ously of their time and effort. This group included a 
number of our Canadian and British allies. In addi¬ 
tion to the assistance given the war effort, a consider¬ 
able contribution has been made to the knowledge of 
short-wave transmission and especially to the inter¬ 
relation of this phenomenon with meteorological 
conditions. Such information will be most valuable 
in weather forecasting and in furthering the useful¬ 
ness of the whole radio field. 

Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. Conant, Chairman 
National Defense Research Committee 


v 





















































































































































































































































FOREWORD 


T he success of the propagation program was the 
result of the wholehearted cooperation of many 
individuals in the various organizations concerned, 
not only in this country but in England, Canada, 
New Zealand, and Australia. The magnitude of the 
research work accomplished was possible only 
because of the willingness of the workers in many 
organizations to undertake their parts of the overall 
program. In fact, the entire program of the Com¬ 
mittee on Propagation was carried out without the 
necessity of the Committee exercising directive 
authority over any project. 

Dr. Hubert Hopkins of the National Pl^sical 
Laboratory in England and Mr. Donald E. Kerr of 
the Radiation Laboratory at the Massachusetts 
Institute of Technology, who were working on this 
phase of the war effort when the Propagation Com¬ 
mittee was formed, were instrumental in giving a 
good start to its activities. The largest single group 
working for the Committee was under Mr. Kerr. 

The existence of a common program for the 
United Nations in radio-wave propagation resulted 
from the splendid cooperation given the Propagation 
Mission to England by Sir Edward Appleton and 
his Ultra Short Wave Panel. Later, through the 
cooperation of Canadian engineers and scientists, 
Dr. W. R. McKinley of the National Research 
Council of Canada and Dr. Andrew Thomson of the 
Air Services Meteorological Division, Department 
of Transport, Toronto, Canada, undertook to carry 
on a part of the program originally assigned to the 
United States. The program was further rounded 
out by the willingness of the New Zealand Govern¬ 
ment to undertake an experiment for which their 
situation was particularly favorable. Dr. F. E. S. 
Alexander of New Zealand and Dr. Paul A. Anderson 
of the State College of Washington initiated this 
work. Needless to say, the labor of the Committee 
on Propagation could hardly have been effective 
without the cooperation of the Army and Navy. 
Maj. Gen. H. M. McClelland personally established 


Army cooperation, and Lt. Comdr. Ralph A. Krause 
and Capt. Lloyd Berkner were similarly helpful in 
organizing Navy liaison and help. 

Officers and scientific workers of the U. S. Navy 
Radio and Sound Laboratory at San Diego, Cali¬ 
fornia, altered their program on propagation to fit 
in with the overall program of the Committee. 
Capt. David R. Hull, Bureau of Ships, understand¬ 
ing the importance of the technical problems, paved 
the way for effective cooperation by this laboratory. 

Dr. Ralph Bown, Radio and Television Research 
Director, Bell Telephone Laboratories, integrated 
the research programs undertaken by Bell Telephone 
Laboratories for the Committee on Propagation. 
This joint research program included meteorological 
measurements on Bell Telephone Laboratories 
property by meteorologists of the Army Air Forces 
working with Col. D. N. Yates, Director, and 
Lt. Col. Harry Wexler of the Weather Wing, Army 
Air Forces. The accomplishments of the Committee 
on Propagation are a good example of the effective¬ 
ness of cooperation — all parts were essential and 
none more than the rest. 

I want to thank Dr. Karl T. Compton, President 
of Massachusetts Institute of Technology, who was 
always willing to discuss problems of the Committee 
and who helped me to solve many of the more 
difficult ones, and also, Prof. S. S. Attwood, Uni¬ 
versity of Michigan, whose continual counsel 
throughout my term of office was in no small way 
responsible for the success of our activity. 

Credit is also due Bell Telephone Laboratories, 
which made my services available to the Govern¬ 
ment and paid my salary from August 1943 to 
September 1945, and to Cornell University, which 
has allowed me time off with pay to complete the 
work of the Committee on Propagation since 
September 1945. 

Chas. R. Burrows 
Chairman, Committee on Propagation 










PREFACE 


T he material presented in this book was pre¬ 
pared by the Columbia University Wave 
Propagation Group at the request of the Committee 
on Propagation of the National Defense Research 
Committee. The International Radio Propagation 
Conference, meeting at Washington in May 1944, 
recommended that a book be prepared dealing with 
problems of radio wave propagation in the standard 
atmosphere at frequencies above 30 megacycles. 
The importance of these higher frequencies is 
apparent when it is recalled that most radars operate 
in this range and that an increasing number of 
communication circuits are being equipped for 
operation above this frequency. 

A certain amount of evidence from operational 
theaters indicates that lack of familiarity with the 
underlying theory of propagation and calculations 
based thereon not infrequently has resulted in 
ineffective installation and operation of radar and 
communication sets. This is ascribable, in part at 
least, to the lack of publications which give a clear 
picture of the problems of propagation or show how 
the important factors may be evaluated. 

A considerable volume of basic material on 
propagation had appeared in technical journals 
before World War II, and during it a great quantity 
of classified material has come from laboratories and 
operational theaters illustrating new applications of 
old principles, giving valuable information on propa¬ 
gation problems as well as on characteristics of 
radar and communication sets, antennas, targets, 
siting problems, etc. But this information has not 
been coordinated under one cover for practical use 
by signal personnel in the field. The Columbia 
University Wave Propagation Group was asked to 
undertake this task and it is hoped that this book 
will, in some measure, answer the need. 

Our effort, then, has been to provide a book de¬ 


signed for men with college training in radio, physics, 
or electrical engineering, which states the basic laws 
of propagation, that is, shows how the characteristics 
of the earth and the atmosphere control the propaga¬ 
tion of radio waves; gives the fundamental properties 
of the basic types of antenna systems, particularly 
their directivity and gain; gives the reflecting 
properties of targets such as airplanes for use in 
detection by radar; teaches the reader how to calcu¬ 
late field strength or obtain the coverage diagrams 
given a particular set, power, and site; gives the 
fundamental information required in the above 
calculations for application to the radar and com¬ 
munication sets used in operational theaters; and 
provides illustrative material and sample calcula¬ 
tions which show how the laws of propagation may 
advantageously be used in the location and operation 
of radar systems, communication sets, and counter¬ 
measure equipment designed to deceive the enemy 
and to prevent jamming of equipment by enemy 
action or by mutual interaction of our own sets. 

The members of the group chiefly responsible for 
the preparation of the manuscript were Drs. S. 
Rosseland, W. M. Elsasser, P. Newman, and 
Prof. S. Fich. Others who contributed special sec¬ 
tions were Messrs. E. R. Wicher, M. Ettenberg, 
M. Siegel, and Capt. E. J. Emmerling, on special 
assignment from the Signal Corps. 

The editor wishes to acknowledge also the courtes3 r 
of the Radiation Laboratory Wave Propagation 
Group, under Mr. Donald E. Kerr, in supplying the 
universal coverage charts given in Chapter 6, and 
the steady interest and assistance rendered by 
Dr. Chas. R. Burrows, Chairman, NDRC Com¬ 
mittee on Propagation. 

Stephen S. Attwood 
Editor 





































































































































































4 * 

JTKhi ' 






















































CONTENTS 


CHAPTER PAGE 

1 Propagation of Radio Waves: Introduction 

and Objectives. 1 

2 Fundamental Relations.12 

3 Antennas.22 

4 Factors Influencing Transmission ... 45 

5 Calculation of Radio Gain.60 

6 Coverage Diagrams.129 

7 Propagation Aspects of Equipment Operation 160 

8 Diffraction by Terrain.170 

9 Targets.182 

10 Siting.187 

Glossary.197 

Bibliography 3 

OSRD Appointees.199 

Contract Numbers.200 

Service Project Numbers.201 

Index.203 


‘Refer to Bibliography of C. P. Summary Technical Report, Volume 1. 























* 











































Chapter 1 


PROPAGATION OF RADIO WAVES: 
INTRODUCTION AND OBJECTIVES 


11 FACTORS INFLUENCING PROPAGATION 

T he propagation of radio waves through the 
atmosphere and around the curve of the earth, 
at frequencies above 30 me, is influenced by so many 
factors that it is desirable to give first an overall 
survey of the problem. This chapter presents the 
problem of propagation in broad perspective in 
contrast with many of the later chapters which are 
devoted to detailed consideration of special phases 
and methods of calculation. 

1,1,1 Assignment 

The International Radio Wave Propagation Con¬ 
ference recommended that a book be prepared deal¬ 
ing with problems of radio wave propagation in the 
standard atmosphere at frequencies above 30 me. 
The importance of these higher frequencies is appar¬ 
ent when it is recalled that most radars operate in 
this range and that an increasing number of com¬ 
munication circuits are being equipped for operation 
above this frequency. 

A certain amount of evidence from operational 
theaters indicates that lack of familiarity with the 
underlying theory of propagation and calculations 
based thereon not infrequently has resulted in inef¬ 
fective installation and operation of radar and com¬ 
munication sets. This is ascribable, in part at least, 
to the lack of publications which give a clear picture 
of the problems of propagation or show how the 
important factors may be evaluated. 

A considerable volume of basic material on propa¬ 
gation had appeared in technical journals before 
World War II, and during the war a great quantity 
of classified material came from laboratories and 
operational theaters illustrating new applications of 
old principles, giving valuable information on propa¬ 
gation problems as well as on characteristics of radar 
and communication sets, antennas, targets, siting 
problems, etc. But this information has not been 
coordinated under one cover for practical use by sig¬ 
nal personnel in the field. The Columbia University 
Wave Propagation Group was asked to undertake 


this task, and it is hoped that this volume will, in 
some measure, answer the need. 

1,1,2 Purpose 

Our effort then has been to provide a book, de¬ 
signed for men with college training in radio, physics, 
or electrical engineering, which: 

1. States the basic laws of propagation, that is, 
shows how the characteristics of the earth and the 
atmosphere control the propagation of radio waves; 

2. Gives the fundamental properties of the basic 
types of antenna systems, particularly their direc¬ 
tivity and gain; 

3. Gives the reflecting properties of targets such 
as airplanes for use in detection by radar; 

4. Teaches the reader how to calculate field 
strength or obtain the coverage diagrams, given a 
particular set, power, and site; 

5. Gives the fundamental information required in 
the above calculations for application to the radar 
and communication sets used in operational theaters; 

6. Provides illustrative material and sample cal¬ 
culations which show how the laws of propagation 
may advantageously be used in the location and 
operation of radar systems, communication sets, and 
countermeasure equipment designed to deceive the 
enemy and to prevent jamming of equipment by 
enemy action or by mutual interaction of our own 
sets. 

12 FUNDAMENTAL PROBLEMS 

AND LIMITATIONS 

1,2,1 Meaning of Propagation 

By propagation is meant the movement of radio 
waves through the atmosphere, and the transfer, by a 
wave mechanism, of radiant energy from a transmit¬ 
ting antenna to a receiving antenna. The problem 
of propagation requires an understanding of the 
manner in which the wave energy is emitted and 
received as well as of the manner in which it flows 
through the atmosphere. The radio engineer must 


1 


2 


PROPAGATION OF RADIO WAVES 


understand this general mechanism, be able to eval¬ 
uate the factors which play contributory roles, and, 
for a given amount of power emitted from a given 
transmitter, be able to compute the strength of the 
radiation field at any point in space or to locate all 
the points in space where a given field strength 
occurs. 

The problem divides naturally into two parts, (1) 
the one-way transmission or communication problem, 
and (2) the two-way transmission or radar problem. 
In the former the prime requisite is to calculate the 
amount by which the wave and its field strength are 
attenuated in passing from the transmitter to a 
receiver and yet permit a field at the receiver suffi- 


122 Atmospheric Layers 

The atmosphere from one point of view is com¬ 
posed of two layers, the troposphere and the strato¬ 
sphere. The former is a layer adjacent to the earth 
which extends upward approximately 10 km, in 
which the temperature decreases about 6.5 C per 
kilometer with increasing altitude to a value, at the 
upper boundary, of about — 50 C. Above this is 
the stratosphere in which the temperature remains 
approximately constant at — 50 C. 

The ionosphere, as its name implies, is a layer (or 
series of layers) composed of ions and free electrons 
lying at an elevation of approximately 100 km. See 



Figure 1. Transmission along and reflection from the ionosphere occurs primarily with frequencies below 30 me. 
At higher frequencies useful transmission is primarily concerned with the nearly horizontal rays in the troposphere; 
higher angle radiation passes through the ionosphere and is lost. 


cient at least to produce the minimum detectable 
signal. In the latter problem the attenuation must 
be calculable for the two-way journey from trans¬ 
mitter to the target and back to the receiver, which 
frequently uses the same antenna as the transmitter. 
In this type of problem, due account must also be 
taken of the reflecting and reradiating properties of 
the target. 

Knowledge of these factors is indispensable for the 
design, installation, and successful operation of both 
communic ition and radar systems. 


Figure 1. These layers play an important role in the 
transmission of waves at frequencies below 30 me 
and are responsible for transmission over very long 
distances. At the higher frequencies, which are the 
concern of this volume, the portion of the waves 
which penetrate the ionosphere is not useful for 
transmission. 

From this it follows that propagation at the higher 
frequencies (above 30 me), to be useful, must occur 
entirely in the troposphere. This volume therefore 
is concerned only with tropospheric propagation. 






FUNDAMENTAL PROBLEMS AND LIMITATIONS 


3 


12 3 Standard Atmosphere 

Propagation of radio waves in the troposphere is 
materially influenced by the distributions of tem¬ 
perature, pressure, and water vapor. The condition 
most nearly approximated in the Temperate Zone has 
been accepted as the so-called standard atmosphere, 
and propagation under this condition has been stud¬ 
ied and calculations made thereon. 

In the standard atmosphere specified by the 
National Advisory Committee on Aeronautics 
[NACA] the temperature is assumed to decrease with 
altitude at the rate of 6.5 C per kilometer, starting 
from 15 C at sea level, with a sea level dry air 
pressure of 1013.2 millibars, which is equivalent to 
760 mm Hg pressure (see Table 1). 


the total pressure and moisture vapor pressure, re¬ 
spectively, in millibars, at height h above sea level. 
In the moist standard atmosphere, n decreases lin¬ 
early with height h at the approximate rate of 
0.039 X 10 -6 units per meter. 

There are several reasons why this book concerns 
propagation in the moist standard atmosphere. 

1. The atmosphere in certain places (particularly 
the temperate zones) and over considerable periods 
of time is substantially standard in character. 

2. Calculations based on the standard atmosphere 
serve as a standard against which propagation in 
nonstandard atmospheres may be compared. 

3. A great deal of propagation information now 
available in the field is based on propagation cal¬ 
culated for standard conditions. 


Table 1. Properties of the atmosphere. 


NACA standard dry atmosphere 


Moist standard atmosphere 


h 

Altitude 

t 

Temp 

C 

p — e 

Dry air 
pressure 
millibars 

Index of 
refraction 
(n - 1)10 6 

e 

Water vapor 
pressure 
millibars 

Saturated 
water vapor 
pressure 
millibars 

Per cent 
relative 
humidity 

Index of 
refraction 
(n - 1)10* 

Feet 

Meters 

0 

0 

15.0 

1,013.2 

278 

10 

17.1 

58.5 

318 

500 

152 

14.0 

995 

274 

9.5 

16 

59.4 

312.4 

1,000 

305 

13.0 

977.1 

270 

9 

15 

60.0 

309 

1,500 

457 

12.0 

960 

266 

8.5 

14 

60.7 

304 

2,000 

610 

11.0 

942.1 

262 

8 

13.1 

61 

295.6 

3,000 

915 

9.1 

908.1 

254 

7 

11.6 

60.3 

284 

4,000 

1,220 

7.1 

875.1 

247 

6 

10.1 

59.4 

273 

5,000 

1,525 

5.1 

843 

240 

5 

8.8 

57 

262 


To simulate the actual atmosphere of the temper¬ 
ate zones it is necessary further to specify a water 
vapor pressure. The value chosen is 10 millibars at 
sea level, decreasing with altitude at the rate of 
1 millibar per 1,000 ft up to 10,000 ft. With this 
addition the conditions for a moist standard atmos¬ 
phere are specified in Table 1. This value of water 
vapor pressure yields a value of relative humidity 
approximating 60 per cent. 

Listed also in Table 1 is the index of refraction n. 
The gradient of this quantity, dn/dh, controls the 
curvature of the rays for a wave moving in the ap¬ 
proximately horizontal direction; n is given by the 
formula 

(n — 1)10 6 = — yp — e -\ J, (1) 
where T is the absolute temperature, p and e are 


1.2.4 Propagation in the Moist 
Standard Atmosphere 

The radiation energy emitted by a transmitter is 
a wave spreading out in three dimensions, which 
may be represented by a series of concentric spherical 
wave fronts or by a system of lines called rays. The 
velocity at any point on the wave front is given by 

c 3 X 10 8 . , , 0 x 

v = — =- meters per second. (2) 

n n 

Since n decreases with height, the upper portions of 
the wave front move with higher velocities than the 
lower portions, and the wave paths as represented 
by the rays are curved slightly downward, as shown 
in Figure 2. 

The radius of curvature of the rays p is given by 




















4 


PROPAGATION OF RADIO WAVES 


— ^ = -f 0.039 X 10’ 6 units per meter (3) 

p dh 

and p, therefore, is equal to 25.5 X 10 6 meters, which 
is approximately four times the radius of the earth 
(p = 4a). As a result the distance to the radio horizon 
is some 15 per cent greater than the geometrical line- 
of-sight distance from the transmitter to the horizon. 
This curvature of the rays by the atmosphere is 
called refraction. 



Figure 2. Curvature of rays in the standard atmos¬ 
phere. 


For the purpose of calculating wave propagation, 
only relative curvature of the earth and the rays is 
of interest. We can be compensated for the effects 
of refraction by replacing the actual earth with a 
radius a by an equivalent earth with a radius ka 
and replacing the actual atmosphere (in which the 
index of refraction n decreases with height) by a 
homogeneous atmosphere (constant n) in Avhich the 
rays are straight lines. Since 1/a is the curvature 
of the earth and 1/p the curvature of the rays, we 
may set their difference equal to l//ca, the curvature 
of the equivalent earth. Thus 

\ _ 1 = J_ 

a p ka 

and (4) 


1 -(«/p) 1 + «S 

Since p = 4a, k = 4/3, and ka, the radius of the 
equivalent earth, equals 8.49 X 10 6 meters. See 
Figures 4 and 5 in Chapter 4. 

12 5 Propagation in Nonstandard 
Atmospheres 

Though this subject is beyond the scope of this 
volume it is desirable to present a brief discussion 
of the salient features. 


Not infrequently the lower atmosphere is stratified 
in horizontal layers in which the variations with 
height of the temperature and moisture content 
are nonstandard. Of particular interest is a sharp 
rise in temperature with increasing height (tempera¬ 
ture inversion), or a sharp decrease in water vapor 
content, or a combination of the two. If these varia¬ 
tions from the standard distribution are sufficiently 
great, horizontal radio ducts may be formed in the 
atmosphere. In this event an appreciable fraction 
of the wave energy (only that fraction moving in the 
nearly horizontal direction) may be constrained to 
propagate along the duct to distances far beyond 
the horizon and the field strength may be large 
compared with that obtainable under standard 
conditions. This phenomenon produces a marked 
bending of the wave paths or rays and is known as 
super-refraction. To take fullest advantage of this 
phenomenon the antennas should be located in the 
duct. 

Ducts are of various types: 

1. Overland. These are surface ducts formed at 
night by the radiation cooling of the earth. 

2. Oversea. In the trade-wind belt there appears 
to be a continuous duct of the order of 50 ft thick 
starting at sea level. 

3. Land to sea. Warm dry air flowing from land 
out over cooler water often yields surface ducts 
100 or more feet thick. 

4. Elevated. Caused by subsidence of large air 
masses, these ducts may be found at elevations of 
perhaps 1,000 to 5,000 ft and may vary in thickness 
from a few feet to 1,000 ft. They are common in 
Southern California and certain areas in the Pacific. 

Depending upon the strength and the thickness 
of the duct, there is a limiting frequency below 
which the duct cannot trap the wave energy. 
Though trapping does at times occur at 200 me, 
it is more likely to occur at the higher frequencies 
such as 3,000 me. 

Ability to calculate performance under standard 
conditions is necessary if performance under non¬ 
standard conditions is to be evaluated. 

12,6 Radio Gain 

The basic problem to be solved is that of com¬ 
puting the radio gain of a transmitting-receiving 
system. 

The radio gain of a transmitting-receiving system 












FUNDAMENTAL PROBLEMS AND LIMITATIONS 


0 


is defined as the ratio of received power P 2 , delivered 
to a load matched to the receiving antenna, to 
power P\, supplied to the transmitting antenna, with 
both antennas adjusted for maximum power transfer. 
Thus 


Radio gain = 


P2 

Pi’ 


( 5 ) 


which is equal, in the decibel scale, to 
Po 

10 logio — = radio gain in decibels. (6) 

The attenuation is the reciprocal of the gain. Since 
Pi/Pi < 1, the gain in decibels is necessarily a nega¬ 
tive quantity. The attenuation in decibels is a 
positive quantity equal in magnitude to the gain in 
decibels. 

The radio gain can be taken as the product of 
physically significant factors. Among these are the 
gains Gi and (7 2 of the transmitting and receiving 
antennas respectively; and A 2 which accounts for all 
other influences modifying the transmission of power. 
A is called the gain factor. 

Radio gain = — — GiG^A 2 . (7a) 


adjusted for maximum power transfer. A 0 = 3X/87T d 
where d is the distance between doublets. A p is the 
path gain factor which includes all additional influ¬ 
ences modifying the transmission of power. 

These factors may also be related to the field 
strength, E, at any point in space by 

E = Ej0[A p 

and 

A_ = E 

Ao £ 0 V<?, ' 

Here E 0 is the free-space field at a point in space 
set up by a doublet transmitter and E 0 ^Gi is the 
free-space field of a transmitter with antenna 
gain G\. 

The primary function of this book is to show how 
the factors A and A P may be calculated, taking into 
account all contributory influences which modify 
their magnitudes. 


(9) 

( 10 ) 


12 7 Radio Gain of Doublet Antennas 
in Free Space 


The radar problem involves double transmission 
over the path as well as the reradiating properties of 
the target, 167 t<t/9X 2 . 


Radar gain = — = GiG^A 4 
Pi 



(7b) 


where <r is the radar cross section of the target and X 
is the wavelength. 


This is the fundamental and simplest case of 
transmission of radiant energy, against which other 
transmitting combinations may be compared. Two 
doublet antennas (for which the gains, by definition, 
are unity) are set up in free space in a manner which 
insures the maximum transfer of power to the re¬ 
ceiver circuit, i.e., the doublets are parallel to each 
other, have a common equatorial plane, and the 



Figure 3. Doublet antennas in free space. 


The gain factor, A, may also be split into two 
factors, so that 

A = A„A f . (8) 

Here A 0 is the free-space gain factor for doublet 
antennas (see Sections 1.2.7, 2.1.3, 2.2.2, and 5.1.2) 


receiver circuit impedance is matched to that of the 
receiving antenna (see Figure 3). Then for free 
space, 


Free-space gain = 


P± 

Pi 


= ^o 2 = 



(ii) 












6 


PROPAGATION OF RADIO WAVES 


and the free-space field strength at distance d from 
the transmitter is 


E 0 


3 V 5 Vp, 

d 


( 12 ) 


Pi is the power radiated by the doublet transmitting 
antenna (see Section 2.1.1). 


13 SURVEY OF PROPAGATION 
1,3,1 Outline 

We will first consider those factors which are 
instrumental in modifying the transmission or the 
attenuation that arises from the presence of the 
earth, then give typical curves of both vertical and 
horizontal variation of field strength, and lastly, 
consider the problem of coverage. 


1,3,2 Factors Modifying Transmission 

The important factors which affect the distribu¬ 
tion of field strength are the following: 

1. Antenna characteristics. For many applications 
the most important feature is the gain which is a 
measure of the directivity of the antenna. From a 


pattern gives the relative amount of power per unit 
area radiated in that direction. 


OO 

Doublet 

Antenna 



Figure 4. Antenna radiation patterns. 

2. Polarization. The wave is said to be polarized 
horizontally or vertically according to whether the 
electric vector E is parallel to the earth’s surface or 
is in a plane perpendicular thereto. A horizontal 
electric doublet (axis parallel to the earth’s surface) 
radiates horizontally polarized waves, whereas a 
vertical doublet radiates vertically polarized waves. 

Too many factors are involved to make it possible 
to state in general which type of polarization should 
be used in a particular case. 

3. Refraction. As explained in Section 1.2.4, 
refraction in the standard atmosphere can be taken 
into account by using an equivalent earth with a 
radius equal to */$ that of the actual earth and a 
homogeneous atmosphere in which the rays traverse 
straight line paths. 


Transmitter 


Receiver 



value of 1.09 for the half-wave dipole, the gain may 
increase to several thousand for the highly directive 
parabolic antennas used in the microwave range. 

Antennas with high gains which concentrate the 
radiated energy into beams of small angles require 
less power to produce a detectable signal. This is 
particularly important in radar where the attenua¬ 
tion of the two-way path is pronounced. 

Qualitative radiation patterns for the doublet 
antenna and an antenna with high directivity are 
illustrated in Figure 4. The radial distance to the 


4. Reflection. Well above the line of sight (see 
Figure 5) the field at the receiver R is the vector sum 
of the fields radiated along the paths of the direct 
and reflected rays. The contribution from the 
reflected ray path depends primarily on the manner 
in which the earth (or sea) acts as a reflecting body. 

Upon reflection, the angle of incidence (90° — ^) 
is equal to the angle of reflection, irrespective of the 
polarization of the wave, but the strength of the 
field in the reflected ray relative to that in the inci¬ 
dent ray depends upon (a) the grazing angle \f/, 









SURVEY OF PROPAGATION 


7 


(b) the type of polarization, (c) the reflecting proper¬ 
ties of the earth or sea, and (d) the divergence factor. 

The incident beam or bundle of rays, in general, is 
partially absorbed by the earth, while the reflected 
portion is reduced in strength and suffers a phase 
shift relative to the incident beam. (In the case of 
sea water with horizontal polarization, the earth 
acts substantially as a perfect reflector, for which 
the reflection is 100 per cent complete and the phase 
shift is 180 degrees for all grazing angles. This is 
true for vertical polarization only at zero grazing 
angle.) 

The divergence factor is introduced to account for 
the fact that an incident bundle of rays striking a 
spherical surface diverges upon reflection and pro¬ 
duces a further decrease in strength of the reflected 
beam. 

Reflection from hills, trees, and other obstacles 
must frequently be taken into account, particularly 
in the siting of very high frequency [VHF] com¬ 
munication sets. 

5. Diffraction. The mechanism by which radio 
waves curve around edges and penetrate into the 
shadow region behind an opaque obstacle is called 
diffraction. The explanation usually given is based 
on Huyghens’ principle. This, in effect, states that 
every elementary area on a wavefront (see P in 
Figure 6) is a center which radiates in all directions 



on the forward side of the wavefront; the intensity 
of radiation is a maximum in the direction per¬ 
pendicular to the wavefront and depends on angle 
0 according to the function (1 -f- cos 0). The field at 
any point, either inside or outside the shadow zone, is 
obtained by summing the contributions from all the 
elementary areas comprising the wavefront. 

As a result of these calculations, the field along the 


line A A' in Figure 6 varies approximately as indi¬ 
cated in the curve. Unity represents the field value 
if the obstacle were removed. It is seen that the 
field strength rises from a minimum at point A to 0.5 
at the edge of the shadow zone and thereafter oscil¬ 
lates about unity. The field outside the shadow zone, 
therefore, at certain points is stronger and at other 
points is weaker than it would be if there were no 
obstacle. The curve, of course, varies with the 
position of the line A A', the size and shape of the 
obstacle, the wavelength of the radiation, and the 
type of polarization. The diffraction of radiant 
energy into the shadow zone increases with increas¬ 
ing wavelength. 

Of prime importance for propagation of radio 
waves is the diffraction of these waves into the 
diffraction region below the line of sight (see Figure 
5). But it should be noted that the influence of 
diffraction is not confined to this region but extends 
well above the line of sight. [In general, the influ¬ 
ence extends upward far enough to affect the shape 
of the lower part of the first lobe in a coverage 
diagram (see Figures 25 and 26 of Chapter 5). In 
this region the diffraction contribution must be 
added to the contributions of the direct and reflected 
rays to give the correct value of the field strength 
at R in Figure 5.] 

Of importance in communication problems is 
diffraction of waves around obstacles such as hills, 
trees, houses, etc. This is illustrated in Figure 6. 
Again diffraction is important in problems involving 
propagation above two different earth conditions. 
An especially important case is that of a radar set 
well inland and searching far out over the sea. Here 
the shore line is treated as a diffracting edge for the 
radiation from the image antenna. 

6. Absorption and scattering. No account is taken 
in this book of the absorption and scattering of radio 
waves by the various constituents of the atmos¬ 
phere. Oxygen, water vapor, water droplets, and 
rain all contribute to absorption. Their influence, 
however, is important only in the microwave range 
and in general tends to increase with frequency. 

1,3,3 General Nature 

of the Radiation Field 

In Section 1.3.2, reference has been made to the 
role of reflection by the earth. The resultant of the 
direct and indirect rays at points in the region above 
the line of sight gives rise to the lobes of an inter- 



8 


PROPAGATION OF RADIO WAVES 


ference pattern (see Figures 9 to 12). The maximum 
number of lobes is the largest integral number of 
times that the quarter wavelength is contained in the 
transmitter height. 

In the case of horizontal polarization over a 
smooth surface, e.g., a calm sea, the reflected and 
direct rays are comparable in strength, so that at 
certain points (on lines for which the points corre¬ 
spond to a path difference of a half wavelength) 
where the reinforcement is a maximum, the field 
may be as much as twice the free-space field. More 
exactly, the free-space field is multiplied at points of 
maxima by (1 + FD), where D is the value of the 
divergence factor for the point and F gives the rela¬ 
tive strength of the reflected and direct rays attribut¬ 
able to the antenna beam pattern. At points of 
minima (the nulls) the field is (1 — FD) times the 
free-space field. 

In general the magnitude of the reflected wave is 
reduced both by the increased divergence resulting 
from reflection from the convex surface of the earth 
but also because the electrical properties of the 
earth are such that only part of the incident energy 
is reflected. The magnitude of the reflection coeffi¬ 
cient is then pD instead of the D used in the preced¬ 
ing paragraph, where p is the magnitude of the 
reflection coefficient for plane waves impinging on 
a plane surface. The field strength, then, lies be¬ 
tween (1 + pFD ) and (1 — pFD). As a result of 
the smaller value of pFD for vertical polarization 
the maxima of the interference pattern are reduced 
and the nulls strengthened. 

At low heights (see Figure 3 of Chapter 5) the 
effect of diffraction is important, so that when refer¬ 
ence is made to the optical interference region, it 
should be understood that the portion of the optical 
region near the earth is not included. It must be 
considered instead as part of the diffraction region. 

The diffraction region, accordingly, designates a 
layer in the optical region asVell as the region below 
the line of sight (see Figure 3 in Chapter 5). Below 
the line of sight the field falls off exponentially. 
Within the diffraction region, fields are strengthened 
by raising the receiver or transmitter antennas. 


I.3. 4 Typical Radio Gain Curves 

Three types of graphical representation of radio 
gain in a vertical plane through the transmitter 
antenna are possible, namely, (1) at a specified 
distance, radio gain against height; (2) at a specified 


height, radio gain against distance; and (3) a set of 
contour lines representing constant radio gain. 

10,000 


2 

lit 

£ 1000 

z 

£ 

x 

S3 

bl 

X 

£ ioo 


10 

-240 -220 -200 -180 -160 -140 -120 -100 -80 

20 LOG A IN DB 

Figure 7. Radio gain vs receiver height for horizontal 
polarization. 

In Figure 7, curves of type (1) are exhibited for 
various frequencies. The transmission is over sea 
water with horizontally polarized waves. It may be 
observed that the higher the frequency the lower the 
first maximum and the narrower the lobe. Figure 8 
gives similar information for vertically polarized 


FRE 
— 3 
—1C 

HORIZOf 

TRANS* 

DIS1 

SEA 

OUENCIES 
000 MC 
)0,200,5C 

UAL POL 
UTTER H 
ANCE 8< 
. WATER 

LINE ( 

>OMC 

.ARIZATIC 
IEIGHT 9 
D KM 

1 

)F SIGH! 

)N 

_5( 

L_ - 

_ i 

"D 

200,/ 

100 

M 


J/y 







7 // 




300 

/ 

! 

. 

/ 

/ 

/ 

0/ 

/ 

500 

\ 

! 

DO 





-,- 

FREQUENCIES 

- 3000MC 

-100,200,500 

VERTICAL POLARIZAT 
TRANSMITTER HEIGH1 
DISTANCE 80 KM 
SEA WATER 

MC 

ION 
r 9 M 


JT 

XX 

f 500 

>00 

\ y/ 100 


—LINE C 

IF SIGHT 


/ 

y// 




✓ 

\ 

\ 

\ 

\ 

\ 

/ / 




✓ 

3000/ 

✓ 

✓ 

- 1 - 

/ 

s 

/ 

/ 

/ 

500 

’ 2.00 j 

7 

/IOO 





-240 -200 -180 -160 -140 -120 -100 -80 


20 LOG A IN DB 


Figure 8. Radio gain vs receiver height for vertical 
polarization. 

waves. Note that the minima are not so deep with 
vertical polarization. Curves of type (2) exhibit 
similar characteristics (see Figure 6 of Chapter 5). 

In Figures 9 to 12 a , vertical coverage diagrams of 

a Figures 7 to 12 have been adapted from Radiation Lab¬ 
oratory Report C-6. 



































SURVEY OF PROPAGATION 


9 


type (3) are given. These illustrate the effects of 
frequency, polarization, and transmitter height. 

A comparison of Figures 9 and 10 shows the effect 
of frequency. As the frequency increases, the lobes 
become more numerous, narrower, and lower. 
Another effect is exhibited along the surface. For 
the higher frequency the corresponding decibel lines 
come in closer to the transmitter. This illustrates the 
fact that for the higher frequency the shadow effect 
is more pronounced along the surface of the earth. 

A comparison of Figures 10 and 11 shows that 
for horizontal polarization the nulls are deeper but 
the lobes extend out farther. Along the line of sight, 
vertical polarization gives the higher field strength, 
while well within the diffraction region the field 


strength is about the same. The last observation 
holds for all frequencies greater than 300 me, the 
greater the frequency the less difference in the 
diffraction region between the two polarizations. 

A comparison of Figures 9 and 12 shows the effect 
of the height of the transmitting antenna. As the 
antenna height is increased, the lobes are narrower 
and depressed toward the horizon. The range is 
improved. However, there are broad nulls for the 
higher antenna in which detection will fail. Below 
the horizon, the corresponding decibel contours are 
pushed to the right so that point-to-point com¬ 
munication is improved. It should be observed that 
the effect of height upon the lobe structure is similar 
to that of frequency. 


CD 

O 



Figure 9. Contours of constant radio gain factor for horizontal polarization on 100 me over sea water. 



Figure 10. Contours of constant radio gain factor for horizontal polarization on 3000 me over sea water. 


































10 


PROPAGATION OF RADIO WAVES 


Which contour actually represents the limit of 
detection for a given radar depends on the power 
output of the transmitter, the minimum power 
detectable by the receiver, the antenna gains, and 
the radar cross section of the target. For com¬ 
munication sets, the same quantities, except the 
target cross section, apply. 

14 ORGANIZATION OF THIS VOLUME 

1,4,1 Arrangement of Material 

This volume is composed of two classes of material. 
One class, comprising Chapters 2, 5, and 6, is de¬ 
voted primarily to the major problem of calculating 


the field strength, while the other class discusses 
collateral problems of importance if these calcula¬ 
tions are to be utilized for obtaining the most 
effective use of radar and communication sets. 

Chapter 2 is devoted to a presentation of basic 
relationships such as the definition of radio gain, 
the transfer of power between doublet antennas in 
free space, antenna gain, receiver sensitivity and 
noise, and the definitions of radar gain and cross 
section. 

The problem of computing the field strength or 
radio gain at any point in the atmosphere is given at 
length in Chapter 5, and Chapter 6 extends this 
material to the calculation of coverage diagrams. 
In Chapter 7 these calculations are related to the 



Figure 11. Contours of constant radio gain factor for vertical polarization on 3000 me over sea water. 



Figure 12. Contours of constant radio gain factor for horizontal polarization on 100 me over sea water. 











































UNITS AND FREQUENCY RANGES 


11 


performance characteristics of particular radar and 
communication sets. 

In Chapter 3 will be found a discussion of the 
radiating properties of a wide variety of antennas. 
Particular attention is devoted to a consideration of 
the shape of the radiation patterns, methods for 
improving directivity of antennas, and computation 
of the gains. 

A general discussion of the factors which modify 
the manner in which radio waves are transmitted 
through the atmosphere is given in Chapter 4. Here 
also is given the reflecting properties of sea water 
and various types of soil. 

An important practical problem is that of diffrac¬ 
tion of waves around obstacles such as hills and trees. 
A simplified treatment of this problem is given in 
Chapter 8. 

Chapter 9 is devoted to a presentation of the 
reflecting problems of targets and their bearing on 
the operation of radar sets. Successful operation of 
these sets is dependent, in no small measure, on the 
proper choice of siting. Factors bearing on the siting 
problem are evaluated in Chapter 10. 


15 UNITS AND FREQUENCY RANGES 
151 Units 

In this book the units used are those of the mks 
rationalized system, in which distances are ex¬ 
pressed in meters, masses in kilograms, and time in 
seconds; and the formulas have been rationalized 
so that the factor 4 tt appears in equations involving 
point sources, 2tt in equations involving line sources, 
and is generally absent from equations for uniform 
or unidirectional fields. 

The Coulomb formula, for illustration, for point 
sources in classical electrostatic units, 


Similarly, the Coulomb formula for magnetic poles, 


/ = 


mim 2> 

fir 2 


(15) 


with mi and ra 2 in unit poles in the electromagnetic 
system and n equal to the permeability, transforms to 


/ = 


minh 
47T/io \x r r 2 ’ 


(16) 


where mi and m 2 are now given in webers, ix r is the 
permeability relative to that of free space fi Q . 

In the mks rationalized system, the free-space 
values of e 0 and ju 0 must carry the burden of the 
change of units and the inclusion of 4x, and thus 
take on the values 


e 0 = 8.854 • 10 12 = —10 9 farads per meter, (17) 
367r 


= 47r • 10 7 = 1.257 • 10 6 henries per meter. (18) 


With these values, c, the velocity of light in free 
space, is equal to 

c — —*— = 2.998 • 10 8 = 3 • 10 8 meters per second 

V,°M0 (19) 

and the impedance of free space is 


Jt2 = 


376.7 ohms. 


( 20 ) 


This system of units has been chosen because it is 
unified, free from numerical factors required in 
equations using arbitrary choices of units, and has 
been adopted by the International Electrotechnical 
Commission. Since the various Armed Services 
use differing sets of units for their operational 
instructions, it would have been impossible to choose 
any one that would have been satisfactory to all; 
hence the choice of using the only system which is 
generally recognized and scientifically sound. 


1.5.2 Sy m D 0 ] s f or Frequency Ranges 


/ =^ (is) 

er 2 

with / in dynes, qi and g 2 in statcoulombs, r in centi¬ 
meters and € referred to unity in free space, is 
transformed to 

f = (14) 

4 *w 2 

Here force / is given in newtons (1 newton = 10 5 
dynes), qi and q 2 in coulombs, r in meters, e r is the 
dielectric constant relative to that of free space e 0 . 


The following symbolism has been adopted for 
various ranges of frequency. 


Table 2. Symbols for frequency ranges. 


Symbol 

Frequency 

name 

Frequency 

me 

Wavelength 

meters 

LF 

Low 

0.03-0.3 

10,000-1,000 

MF 

Medium 

0.3-3 

1,000-100 

HF 

High 

3-30 

100-10 short waves 

VHF 

Very high 

30-300 

10-1 \ ultra-short 

UHF 

Ultra-high 

300-3,000 

1-0.1 / waves 

SHF 

Super-high 

>3,000 

<.l microwaves 













Chapter 2 

FUNDAMENTAL RELATIONS 


21 THE ELECTRIC DOUBLET 

IN FREE SPACE 

2,1,1 Radiation of an Electric Doublet 

I t is convenient to present the basic relation¬ 
ships of radiation and reception by antennas in 
their simplest form, that of the radiation and recep¬ 
tion of electric doublets in free space. The resulting 
formulas will later be generalized to include other 
types of antennas and their positions relative to the 
earth. 

An electric doublet is a rectilinear antenna, which 
is symmetrical about the point or points of connec¬ 
tion thereto and is so short that its directive proper¬ 
ties are independent of its length. The field of such 
an antenna does not depend on the distribution of 
current along the wire, because the wire is so short 



that there is no phase difference between waves 
reaching a point in space from different portions of 
the wire. In symbols, l « X where l is the length 
of the antenna and X is the wavelength of the radia¬ 
tion. 

To facilitate the analysis of the field of the doublet 
antenna, the spherical polar coordinate system 
shown in Figure 1 is introduced. The upper half 
of the doublet is shown in the figure. The distance 
from the center of the antenna to a point in space 
is here denoted by r. Elsewhere in this volume 
this quantity is written d. 


Let dl be an infinitesimal portion of l, the length of 
the doublet, and let the current in this portion be 
the real part of Ie j{2irct/K \ Let dE r , dEg, dE$ and dH r , 
dH 0 , dH'f, be the components of the electric and 
magnetic field strengths at any point P(r, 6, <j>) due 
to the current in dl. A straightforward solution of 
the fundamental equations of electromagnetic theory 
gives the following values for these components, 
valid at distances large compared with the length of 
the doublet: 


dE r = 607 


dE. 


njui 

Lr 2 2irr 3 J 
= G07r i\l + J - £_1 

LXr 2tt r 2 47r 2 r 3 J 


dl cos 6 e i{2ir/K){ct ~ r) 

volts per meter, 

dl sin ee** w/k){ct ' f) 

volts per meter, (1) 

dE ^ = 0, dH r = 0, dH 0 = 0, 

1 

2 Lx?- 2?rr 2 


> I 


dl sin0e ,(2 ’ A)(c, “ r) 

amperes per meter, 


where 

c = velocity of light = 3 X 10 8 meters per second 

j = V^T, 


and all distances are measured in meters. Electric 
field strengths are in volts per meter and magnetic 
field strengths are in amperes per meter. Unless 
otherwise explicitly stated, the mks rationalized 
units are used throughout this volume. 

Equation (1) can be simplified at once. Since the 
time variation of the field is assumed sinusoidal, 
e j( 2 nct/\) ma y om itt e d. The term e~ 3{2irr/K) gives the 
phase, and it too can be omitted when only the 
amplitude is required. From here on, unless other¬ 
wise stated, it is understood that root-mean-square 
(rms) values will be used for dE r , dE e , dH and 7. 

The field in the neighborhood of the doublet is 
called the induction field and is given by the terms 
in equation (1) which include the highest powers of 
r in the denominators. This field is important when 
mutual effects between closely spaced antennas, or 
antennas and reflectors or directors, are involved. 

The radiation field, of greater interest for most of 
the purposes of this volume and the only important 
field at large distances (r»X), is given by the 


12 








THE ELECTRIC DOUBLET IN FREE SPACE 


13 


terms in equation (1) containing r _1 . Thus the 
radiation field for the element dl of the doublet may 
be written: 


dE 0 
dH * 


SOirldlsind , 

-volts per meter, 

Xr (2) 

Idl sin 6 _ dE 0 

—~-TT" - amperes per meter. 

2Xr 1207r 


The other components are relatively negligible 
except near the antenna or near ground for low 
antennas. The electric field dE e is perpendicular 
to the radius vector r and lies in the r,z plane, and 
the magnetic field dH$ is perpendicular to r and to 
dE e . It will be noted that E/H = 1207T = 376.7 
ohms. This is the impedance of free space in the 
mks rationalized system of units, the ohms of the 
electrical engineer. 

Equation (2) describes the radiation field of a 
differential element of the doublet. To get the 
radiation field of the whole doublet, these equations 
must be integrated over the length l. This gives 

60 7r sin 6 


H# = jEJ 0 /12O7t amperes per meter. 

Equation (3) may be written in exactly the form 
of equation (2) by introducing the effective length, L, 
of an antenna, which is defined as the length that a 
straight wire carrying current constant over its 
length would haye if it produced the same field as 
the antenna in question. Calling the current meas¬ 
ured at the input point /*, 


/ ‘i/2 

Idl 

1/2 


volts per meter, (3) 


L = 



meters, 


(4) 


and hence 


E e 

H+ 


607t/;L sin 6 
Xr 


volts per meter, 


E 0 

120tt 


amperes per meter, 


(5) 


so that equations (5) are the same as equations (2) 

with Iff replacing JIdl. For a short dipole or 

doublet the current varies linearly from at the 
midpoint to zero at each end so that from equation (4) 
L = 1/2 for a doublet. 

The power per unit area, W (that is, the power 
flowing through a unit area normal to the direction 


of propagation), is represented by Poynting’s 
vector and is given by the product E 0 H$ times the 
sine of the angle between E 0 and H This angle is 
90 degrees. Consequently, 

W = EH watts per square meter, 

E 2 

W = ~~ watts per square meter, (6) 

E = Vl207rIF volts per meter. 

To find P, the power output of the doublet, W is 
integrated over a large sphere concentric with the 
source. Using equations (5), 

E 2 d 2 

P — ~z~ watts 
45 

and (7) 

_ 3V5VF 
d ’ 

where d is written in place of r. The subscripts 6 and 
</> have been dropped at this point because the 
E and H referred to in equations (7) are the fields 
in the equatorial plane, where sin 6 = 1. 

As the antenna is part of a circuit, it is often con¬ 
venient to think of the radiated power as being 
dissipated in a fictitious resistance called the radia¬ 
tion resistance , defined by 

R r = — ohms, (8) 

I i 

where P is the radiated power and /,• the rms input 
current. For the doublet, 



where L is the effective length given by equation (4). 


21,2 Reception by an Electric Doublet 

When an electromagnetic wave falls upon an 
antenna, a current is induced in the antenna and 
power is abstracted from the wave. If the antenna 
is connected to a load, the power abstracted is 
dissipated in two ways: (1) by absorption in the 
load (reception), and (2) by reradiation from the 
antenna (scattering). 

In this classification, the power dissipated by the 
antenna itself (due to its ohmic resistance) is ignored 
because this loss is likely to be negligible compared 
with the power dissipated through reradiation. 
Hereafter, power absorbed by the load will be called 
received power and power reradiated by the antenna 








14 


FUNDAMENTAL RELATIONS 


will be called scattered power. The sum of these is 
equal to the power abstracted from the wave. 

The calculation of the received and scattered 
power may be carried out by means of the equivalent 
circuit of Figure 2. In this figure, Z a is the impedance 



of the doublet and Z* is the impedance of the load, 
that is, the impedance connected across the terminals 
of the antenna when it is acting as a receiver. V is 
the voltage generated in the antenna. 

The load is supposed to be tuned, which means that 
the reactance part of Z t is set equal and opposite 
to the reactance part of Z a , so that Z a + Z x = R a 
+ Ri, that is, the total impedance is simply the 
sum of the resistance parts of the impedances of the 
antenna and the load. Hence 


I = 


V 

R a + Ri 


( 10 ) 


gives the current. But P r = RJ 2 is the power ab¬ 
sorbed by the load and hence is equal to 


V*R l 

(Ra + RlY ’ 


( 11 ) 


where P, is called the received pow r er. In the same 
way 


V 2 R a 

(Ra + Ri ) 2 


( 12 ) 


is the power scattered by the doublet. 

It is easy to show that the maximum power is 
delivered to the load if R a = R t . In this, the matched 
load case, 


Pr 



Z!_ 

4 R t m 


(13) 


Now the resistance of the doublet, neglecting its 
low ohmic resistance, is only the radiation resistance 
[equation (9)] and the potential or voltage across 
the terminals is equal to EoL, where E 0 is the field 
strength of the incident plane wave and L is the 


effective length of the doublet. Inserting 
quantities into equation (13), 


Ep 2 ' 3X 2 
1207T St 


these 


(14) 


In these equations it has been assumed that the line 
of the doublet has been oriented parallel to the 
electric vector of the incident wave in order to obtain 
maximum power absorption. 

The factor E 0 2 /120t will be recognized from 
equation (6) as the power per unit area of the inci¬ 
dent wave. The formula thus says that all the power 
crossing an area 3X 2 /87r is received, and that all the 
power crossing an equal area is scattered. The 
area 3X 2 /87r is therefore called the absorption cross 
section or scattering cross section of the matched 
doublet. Since the antenna has been placed parallel 
to the polarization of the incident wave, this is the 
maximum absorption cross section. Moreover, this 
formula holds only when the doublet has been 
matched to its load, and consequently S\ 2 /St is the 
maximum absorption cross section. 

It will be noted, however, that S\ 2 /St is not the 
maximum scattering cross section. This maximum 
is achieved by shorting out the load, that is, setting 
Ri — 0. In this case, 


and 


Ep 2 ' 3X 2 
120x 2t 


(15) 


Hence the scattering cross section of the shorted 
(dummy) doublet is four times the scattering cross 
section of the matched load doublet. 

It should be noted in passing that the cross sections 
introduced here should not be confused with the 
radar cross section which is discussed in Section 2.4. 


2,1,3 Transmission between Doublets 
in Free Space 

Assume that two doublets, one to function as a 
transmitter and the other as a receiver, a distance 
d >> X apart, are adjusted for maximum power 
transfer. This means that the axes of the doublets 
are parallel and lie in their common equatorial plane 
and that each is matched to its connected circuit. 
Then the power radiated by the transmitting doublet, 
from equation (7), is equal to 

P E ° 2d2 , 1A x 

Pi = - watts, (16) 

45 


















10'0 q 0001 


POWER TRANSMISSION. RECIPROCITY 


15 


imliiiilm ilniiliiiilmili111 1 


111 11 i i i i 


d (KILOMETERS) 

Imiliinllliililiilii 11 1 1111liinliiilln11 1 1 1 1 1 11 1 1 i I i l l i 


llllllljlllllllllllllllllllllllllllll 1 I I LI 


u» 




fo 


111111 | 1111 j 111111111111111 rrrr 


^ o 5 o o o o o 

11^ 1 111111 I 1^111111 11 1^111 111111^1111 [ 11 11 1111 1 1111 1111111 n i 1111 n 11 M 1111111111 1 1 II 

GAIN IN DECIBELS 20 LOG q^-= 20 LOG A 0 


O P O 

b o o 

u» ^ tn 


O p P 

oj ^ cn 


to 


CP 


I I | I | |T T I | I I l l |H I I[l lll | l lll| llll|lll l [IIMj ll l l | lll l| lll l| I I I I | I TT I j I I I I [III Ijm l|llll| I I 11 |ll ll|llll|llll|llll|llll| I I I I j I I I Ijll ll|llll|llli|llll| II 

X(METERS) 

Figure 3. Free-space gain for doublets P 2 /Pi = (3X/87Td) 2 = A 2 0 . (Adjusted for maximum power transfer.) 


and from equation (14) the power delivered to the 
load circuit of the receiving doublet is given by 


P 2 


Po 2 

120 7T 


— watts. 
87r 


(17) 


Hence the ratio of the received power (to the load 
circuit) to the output power for maximum power 
transfer is 



The ratio P 2 /P 1 = A 0 2 (as used here) is called the 
free-space radio gain for matched doublets or for 
short the free-space gain, since all objects, including 
the earth, are supposed remote from both doublets. 
This free-space gain, A 0 = 3X/87rd. On the decibel 
scale, it takes the form 

p 

10 logio — = 20 log Ao 

Pi 

= 18.46 — 20 logio - decibels. (19) 
X 


The nomogram, Figure 3, gives a convenient 
means of calculating the free-space gain for doublets 
adjusted for maximum power transfer. 


2 2 POWER TRANSMISSION. RECIPROCITY 

2,2,1 Radio Gain 

The formulas in Section 2.1 apply to doublets in 
free space. This section considers the modifications 


that must be made in the formulas when the re¬ 
striction of free space is removed. In actual trans¬ 
mission problems, ground reflection, reflection from 
elevated layers of the atmosphere, diffraction by 
earth curvature and by obstacles, and refraction by 
the atmosphere must be considered. In Chapters 5, 
6, and 7 special forms of gain are discussed and 
separate gain factors are introduced to take care of 
each effect. For the present a factor that will be 
called the path-gain factor, A P , representing the 
product of all these special factors will be used. 
A P is defined by 

E = EqA p , (20) 

where E is the absolute value of the actual field 
strength and E 0 is the absolute value of the free- 
space field strength that would exist at the same 
distance d from the doublet transmitter in free space. 

Replacing E 0 with E 0 A P , equation (17) for the 
received power, becomes 

p 2 = l . ^ (21) 

120tt 8t r 

while the power output as given by equation (16) 
remains unchanged, so that 

(22) 

replaces equation (18) as the ratio of received 
power to output power for maximum power transfer 
between doublets. The quantity defined by (22) 
is the free-space gain and A is the gain factor. 










16 


FUNDAMENTAL RELATIONS 


The general relation between the input voltage at 
the receiver and the received power is V t = Vp 2 I? z , 
where Ri is the resistance of the receiver load circuit 
(which is equal to the radiation resistance for maxi¬ 
mum power transfer) and V t is the input voltage. 
Hence, using equation (14), 

V t = 0.0178#o\Vfl* volts (23) 


1 .— E 0 L 

~ 7f E Q \ylRi — 9 

87 rV 5 2 


Antenna Gain. Polarization 


The equations of Section 2.1 may be further 
generalized to apply to any type of antenna through 
the introduction of a quantity called the antenna 
gain. The term gain, as applied to an antenna, is a 
measure of the efficiency of the antenna as a radiator 
or receiver as compared with that of a doublet 
antenna, with all antennas located in free space. 

Quantitatively, the gain, Gi, of a directive trans¬ 
mitting antenna is the ratio of the power P/ radiated 
by a doublet antenna to the power Pi radiated by the 
antenna in question to give the same response in a 
distant receiver, with both transmitting antennas 
adjusted for maximum transfer of power. Hence 

P/ 

G ! = g-. (24) 

The gain G 2 of a directive receiving antenna is the 
ratio of the power Pi" radiated by a transmitting 
antenna, which produces a certain response in the 
matched load circuit of a distant doublet receiving 
antenna, to the power Pi radiated by the same 
transmitting antenna to produce the same response 
in the matched load circuit of the receiving antenna 
in question, with both receiving antennas adjusted 
for maximum transfer of power. Hence 


G<> 



(25) 


From the definitions given above it follows that 
for a transmitting and receiving antenna combina¬ 
tion in free space, with gains Gi and (r 2 and adjusted 
for maximum power transfer, the power ratio is 
equal to 

~ = GiG, (//Y = ChGtU, (26) 

Pi \Stt a/ 


where Pi, £1 are the power output and gain of the 
transmitter and P 2 is the power delivered to the 
matched load of a receiving antenna of gain G 2 . 


If the antennas are not in free space, equation (26) 
becomes 

j = A/ = <*<*( AoApY, (27) 

= GiG 2 A 2 , 

where A is the gain factor and A P is the path-gain 
factor. Note that for highly directive antennas A p 
may depend upon the directivity characteristic of 
the antennas, e.g. when the antenna discriminates 
between the direct and reflected waves. 

Since power is proportional to the square of field 
strength, equation ( 20 ), for any transmitting an¬ 
tenna, becomes 

E = E^(hA„. (28) 

In defining gain, the electric doublet is selected 
here as the comparison antenna in place of the iso¬ 
tropic radiator (that is, a hypothetical antenna 
which radiates equally in all directions) which is 
sometimes used in the literature. Since the gain of 
an isotropic radiator relative to a doublet is the 
gain of any antenna referred to an isotropic radiator 
is 3/2 the value referred to a doublet antenna. 

(/(isotropic) = 1.5 (/(doublet). (29) 

The chief objections to the isotropic radiator are 
that it does not occur in practice and cannot be 
produced experimentally, even approximately. 

In experimentally measuring the gain of an an¬ 
tenna, a half-wave dipole is often used as a reference 
antenna. While the gain of a half-wave dipole rela¬ 
tive to a doublet is approximately unity, being 1.09 
for a very thin dipole, it depends somewhat on its 
actual dimensions so that it is better to express the 
experimental gain in terms of the doublet antenna 
even though a longer antenna is used as a reference 
antenna in making the measurements. 

When antennas are oriented so that the directions 
of polarization make an angle 7 with each other 

DIRECTION OF 
MAX RADIATION 

TRANS REC 

1. . n:::z:.xr 

TRANSMITTER DIRECTION OF RECEIVER END VIEW 

MAX RE-RADIATION 

Figure 4. Relation of antenna axes and wave polariza¬ 
tion. 

(while the maxima of their angular patterns still 
point toward each other), the formulas for power 
transfer, equations (18), (gg), and (27), are multi¬ 
plied by a factor cos 2 7 (see Figure 4). 











RECEIVER SENSITIVITY 


17 


The Reciprocity Principle 


Thermal Noise 


So far in this chapter the radiation and reception 
of power by antennas have been treated separately. 
Actually, many of the properties of an antenna are 
the same for either reception or radiation; in partic¬ 
ular, the current distribution, the effective length, 
and the gain are unchanged. The reciprocity prin¬ 
ciple, from which these propositions may be proved, 
may be stated as follows: If an electromotive force 
V, inserted in antenna 1 at a point x h causes a cur¬ 
rent / to flow at a point x 2 in antenna 2, then the 
voltage V applied at x 2 will produce the same cur¬ 
rent I at Xi. 

From this principle the statement of the equiva¬ 
lence of current distribution, effective length, and 
gain follow readily. 

This theorem does not hold when the propagation 
of a wave takes place in an ionized medium in the 
presence of a magnetic field (the ionosphere), but it 
does hold for all cases of transmission discussed in 
this volume. 


23 RECEIVER SENSITIVITY 

The sensitivity of a radio receiver is that charac¬ 
teristic which determines the minimum strength of 
signal input capable of causing a desired value of 
signal output. In high-frequency receivers the 
limiting factor for reception is usually set noise, 
that is, noise produced in the tubes or other ele¬ 
ments, such as crystals, of the receiver itself. At 
frequencies below about 100 me, atmospheric dis¬ 
turbances sometimes exceed the set noise in intensity, 
but at higher frequencies atmospheric static is 
negligible. Man-made noise (automobiles, etc.) 
may be a source of serious trouble, but such inter¬ 
ference can often be eliminated by proper siting. 
Consequently, for high-frequency receivers, sensi¬ 
tivity may be expressed, at least approximately, in 
terms of set noise only. 

Although set noise has an important bearing on 
sensitivity of radar receivers, there are other factors 
which must be considered for this type of equipment. 

There are several types of set noise. Though all 
noise sources in a v T ell-designed receiver are mini¬ 
mized with the exception of the thermal noise whose 
magnitude is independent of equipment construction, 
the total set noise is usually several times the purely 
thermal noise. 


Thermal noise is generated by the random 
(temperature) motion of electrons in a conductor; 
it is, therefore, a universal propert} r of matter and 
independent of the design features of the receiver. 
The rms thermal-noise voltage that appears across 
the terminals of any circuit element is a function of 
the frequency interval (receiver bandwidth) over 
which the noise is averaged; it is given by 

V„ = V4fcTA/. R, (30) 


where R is the resistance across which the noise 
voltage is measured, A/ the bandwidth in cycles 
per second, T the absolute temperature, and k, 
the Boltzmann constant, is equal to 1.38 X 10~ 23 
watt-second per degree. The noise voltage is inde¬ 
pendent of the reactance components in the circuit. 

Consider now, for the purpose of definition, a 
receiver without internal noise, that is, let all the 
noise be generated in the receiving antenna of re¬ 
sistance R a . If Ri designates the load resistance 
(that is, the resistance of the receiver exclusive of its 
antenna), the average noise power delivered to the 
receiver will be 


Vn'Rl 

(Ra+Rl) 2 ’ 


(31) 


where V n is the rms value of the noise generated in 
the antenna. 

The noise power is maximum w r hen the receiver is 
matched to its input; this maximum is 

P n = — = kTAf watts (32) 

4 R a 

by equation (30). Assuming equivalent temperature 
T = 290 degrees absolute, and measuring A/ in 
megacycles, 

P n = 4 X 10 -15 A/ watt. (33) 

This result means that in an idealized receiver, 
noise is the thermal noise of an antenna of equivalent 
temperature T — 290 degrees absolute, and the 
minimum detectable signal would be approximately 
equal to 4 X 10 -15 A/ watt. 


2 * 3 ' 2 Noise Figure 

The sensitivity of a set cannot be described in 
terms of the thermal noise alone, because the set 
noise is usually several times the purely thermal 
noise. For this purpose another quantity called the 





18 


FUNDAMENTAL RELATIONS 


noise figure is used. The noise figure of a system 
(taken here to be a receiver, for definiteness) is 
defined as 


F«- 


P no/Pni 
Pso/Psi ’ 


(34) 


where P w - = noise power ( kTAf) from the antenna 
which is being delivered to the receiver. 

P no = noise power at the output of the re¬ 
ceiver, that is, the noise after the 
amplifications and additions arising in 
the receiver circuit. 

Psi = signal power from the antenna which is 
being delivered to the receiver. 

P$o — signal power at the output of the 
receiver, that is, the signal power after 
detection and amplification have taken 
place. 

The ratio P s «/P s i is called the receiver gain. This 
quantity is called g and must not be confused with 
antenna gain G. Using equation (32), equation (34) 
may be written 


The bandwidth Af is measured by finding the area 
under a curve of power-gain versus frequency and 
•equating this area to the area of a rectangle whose 
width is interpreted as Af and whose height corre¬ 
sponds to the gain at the frequency at which the 
gain is a maximum. 


2,3,3 Receiver Sensitivity 

Frequently receiver sensitivity is defined by the 
assumption that a received signal can be discrim¬ 
inated when its output power is equal to the noise 
output power. This assumption, while true for a 
large class of receivers, is too rough for radar re¬ 
ceivers. The method given here will explain the 
procedure used for calculating the minimum dis¬ 
cernible power of receivers for which the assumption 
is true. The sensitivity of radar receivers is con¬ 
sidered in Section 2.3.5. 

Referring to equation (34), the assumption that 
signal output power is equal to noise output power 
means that P so = P no . Hence 

Fn=^. (36) 

Pni 


But P S i is, on the assumption discussed above, just 
the minimum discernible signal power, P m i n , at 
the receiver input, that is, before amplification. 
Hence, using equations (32) and (33), 

P min = kTAf • = 4 X 1CT 15 A/. F n watt. (37) 


2,3,4 Measurement of the Noise Figure 


Remembering that the cases under discussion are 
those for which the minimum discernible signal is 
equal to the noise output power, equation (37) gives 
an estimate of the minimum detectable pow r er from 
a measurement of the noise figure F n which may be 
obtained as follows. 

An antenna (or other signal generator) v r hose 
impedance is matched to the receiver is connected 
to the receiver. With the signal output reduced to 
zero (so that the antenna furnishes only noise power 
to the receiver), the receiver gain is increased until 
the noise gives a measurable output and the output 
noise power is measured with a power meter. Now 
a signal is impressed on the antenna and increased 
to a point where the receiver output power is doubled, 
and the input signal power is measured. Thus, 
referring to equation (34), 


Pin = Pr, 


and 


p = ^ si = ^ si 

n ” P ni ~ kTAf ’ 


so that the measurement of the impressed signal 
power indicated here gives F n . 

If the receiver consists of several elements in 
cascade, including attenuators, amplifiers, and con¬ 
verters, the overall noise figure can be compounded 
from the noise figures and gains of the individual 
components by means of the following equation: 

F n = F nl + + ——, (38) 

01 0102 

where F n = overall noise figure, 

F nk = noise figure of the kth element, 
g k = gain of the kth. element. 


In using this equation it is understood that the suc¬ 
cessive stages are matched. 

It is clear from equation (38) that most of the 
noise comes from the early stages of reception; in 
high-frequency radar sets, it comes from the crystal 
mixer and the first intermediate-frequency (i-f) 
stage. This means of course that noise picked up at 







RADAR CROSS SECTION AND GAIN 


19 


later stages is much less amplified by the system 
than the noise from the early stages. 

In equipment specifications, the noise figure is 
usually expressed in the decibel scale as decibels 
above thermal noise. Actual noise figures vary from 
a few decibels above thermal noise in the very high- 
frequency [VHF] region (receivers built a few* years 
ago often have appreciably higher noise figures) to 
larger values for microwave receivers. 

2 3,5 Sensitivity of Radar Receivers 

It is by no means true for radar receivers that 
P m in= Pno', as a matter of fact, P m in>>Pno- That 
is, the minimum discernible power considerably 
exceeds the noise level. 

The largest single additional loss in radar recep¬ 
tion is scanning loss which is relate to the rotation 
of the antenna (one or several revolutions per 
minute). As an example, for one particular radar 
which has a bandwidth Af = 2 me, this loss is from 
10 db to 12 db. 

In case the antenna does not rotate, there is no 
scanning loss. This fact would seem to be of limited 
operational importance, since it would usually be 
necessary to locate the target (a plane, for example) 
by scanning. 

Another loss, closely connected with scanning 
loss, is sweep-speed loss. This loss is due to the fact 
that practical targets, such as airplanes, reflect 
rapidly varying amounts of power to the radar 
receiver, these amounts depending on the precise 
orientation of the target at the moment when the 
radar beam sweeps over it. Consequently, sweep- 
speed loss will depend on the speed of rotation of the 
antenna, on the distance of the target from the 
antenna, and, to some extent, on the beamwidth 
and the nature of the target. The overall figure 
for this loss on the same radar used to illustrate 
scanning loss is about 4 db for targets 200 miles 
from the radar. 

In addition to these losses, careful experiments 
with the radar used as an example above have 
indicated that there is an operator loss of about 
4 db for even experienced operators. This might be 
thought of as a loss due to the difference between 
laboratory and field conditions. 

Statistical consideration about the extent of noise 
fluctuation and about the fact that a target need 
not be seen on every sweep lead to further small 


losses which total, for the radar under discussion, 
2 db. 

Summarizing for the case of the radar of the above 
example, the minimum detectable power is about 
34 db above kTAf or about 8 X 10 _1L6 watt, not 
12 db above kTAf or 8 X 10 -13 ' 8 watt, as would be 
indicated from the noise level alone. This amounts 
to 22 db or a factor of 166; that is, the actual min¬ 
imum discernible power is 166 times that calculated 
from noise alone. It will be seen in the results of 
the next section that the maximum range of a radar 
set varies with the inverse fourth root of the min¬ 
imum discernible power. Consequently, a calcula¬ 
tion of the maximum range of the radar of the 
example, which assumed that the minimum dis¬ 
cernible power was equal to the noise power, would 
give a range too great by a factor of n/ 166 = 3.59. 
Since this would be a serious error, it shows the 
importance of a very careful consideration of radar 
receiver sensitivity in calculations of this type. 

24 RADAR CROSS SECTION AND GAIN 
2,41 Radar Cross Section 

The total scattering of a target may be described 
by the use of a parameter (having the dimensions 
of an area) called a scattering cross section. This 
concept has already been presented in the latter part 
of Section 2.1.2, where both scattering and absorp¬ 
tion cross sections of doublets were discussed. The 
scattering cross section S is defined by 



where P s is the total power scattered by the target 
irrespective of its angular distribution and is 
the incident power per unit area. 

The scattering cross section S, which gives in¬ 
formation about the total scattered energy, is not 
directly useful in radar work because in such applica¬ 
tions one is interested only in that fraction of the 
total scattered power which is scattered in the direc¬ 
tion of the radar; that is, one wants a parameter 
involving the scattered power per unit area at the 
receiver instead of the total scattered power. If 
the target is an isotropic scatterer, 




20 


FUNDAMENTAL RELATIONS 


where W r is the scattered power per unit area at the 
receiver, d the distance from the target to the 
receiver, and P s the total scattered power. This 
gives, using equation (39), 

S = 4ird 2 —r (40) 

Wi 

as a formula for the scattering cross section of an 
isotropic scatterer which involves scattered power 
per unit area at the receiver W r instead of total 
scattered power P s . 

For targets other than isotropic scatterers, how¬ 
ever, this procedure fails since one cannot say that 
W r = Ps/^Trd 2 . Nevertheless, it is useful to define a 
parameter <r which is called the radar cross section, by 

W 

<r = 4 Trd 2 — f (41) 

Wi 


in analogy with equation (40). Here W r is the actual 
power per unit area at the receiver. From the pre¬ 
ceding discussion it is apparent that cr may be 
thought of as the scattering cross section which the 
target in question would have if it scattered as much 
energy in all directions as it actually does scatter 
in the direction of the radar receiver. For a target 
scattering isotropically, cr = S, but for any other 
type of target a does not, in general, equal S. 

A radar gain formula analogous to the radio gain 
but applicable to two-way transmission can be de¬ 
veloped from equation (41) by replacing W T and Wi 
with the directly measurable quantities Pi (power 
output) and P 2 (received power). From equation 
(6), Wi = E 2 / 120 7r in which E is the field strength 
incident on the target. Substituting this value of E 
into equation (7) gives Wi = SPi/Srrd 2 for a doublet 
transmitter in free space. Including the gain of any 
type of transmitting antenna, this takes the form 


W, = 


3 PA 

8 ird 2 


(42) 


Further, the power received by a doublet with a 
matched load, equation (17), may be written 


-™w 
8 


n 


(43) 


if P 2 /1207r is replaced by W r , w r here here E is the 
field at the receiver. If the receiver is not a doublet, 
equation (43) may be replaced by 
0\2 

l\ = ^ W r G, (44) 

87r 

where G 2 is the gain of the receiver. Substituting the 


values for W{ and W r , given by equations (42) and 
(43), into equation (41) yields 



This is the radar gain for two-way transmission in 
free space. By means of it, a may be measured, or if 
cr and P2/P1 are known, it may be used to calculate 
ranges. Generalizing equation (45) we have 



where A P is the path gain factor (see Section 2.2.1). 

It may be observed here that some writers call 
<fA p a , not cr, the radar cross section. These writers 
call their cr, for the case A P = 1 (free space), the free- 
space radar cross section o- 0 . Since, in this volume, 
the complicated terms appearing in A p are treated 
separately and not as part of the cross section, this 
distinction is not made here. 

For some simple targets, a may be calculated. 
The following are a few of the values. 


Targets 

Condition 

Radar 

cross 
section cr 

Conducting sphere, radius a 

a > > X 

7ra 2 

Metallic plate, area = ab 

a>> X, b >> \ 

4tt«26VX 2 

Cylinder, diameter = d, 
length = l 

Axis of cylinder 
parallel to field 
and d > > X, 
l>>\ 

7 rdl 2 /\ 

Matched load doublet 

Oriented parallel 
to field 

9\ 2 /167r 

Shorted doublet (dummy) 

Oriented parallel 
to field 

9X74 7T 


Objects of tactical interest (ships, airplanes) have 
very complicated radar cross sections. In particular, 
a strong dependence on the aspect of these unsym- 
metrical targets is observed. For ships the situation 
is still further complicated by the variability of the 
incident field over the target area. 

Some writers on the subject of targets use a 
characteristic length L (sometimes also called a 
scattering coefficient) which is related to a by 

cr = 47rL 2 . (47) 

2 4 2 Radar Gain 

It is possible to write equations for two-way 
transmission which bear a formal resemblance to 
corresponding equations for one-way transmission by 











RADAR CROSS SECTION AND GAIN 


21 


introducing a quantity G R , called the gain of the 
target. G R is the gain of a target in the direction 
of the radar receiver relative to a shorted (dummy) 
doublet. 

By writing formulas connecting the radar gain 
with the power per square meter incident on the 
target and the power per square meter scattered 
back to the receiver, it is possible to establish a 
connection between radar gain and the radar cross 
section defined in the last paragraph, and from this 
to calculate a gain formula involving G R instead of <r. 

Applying equations (15) and ( 6 ), 


p = w 

8 * 2tt 


(48) 


for the case where the target is a shorted doublet. 
P s is the total scattered power and IF,- is the power 
per square meter incident on the target. For a 
target with a radar gain G R it follows that 

e. - w, 


2tt 


G, 


(49) 


In a similar way a formula for W r , the scattered 
power per unit area at the receiver, can be developed. 
A target which scattered equally in all directions 
would scatter an amount 


But 


P/ = 4rfIF r . 


P/ = 5 g r p s , 


(50) 


(51) 


where P s is the amount scattered by an actual 
target with gain G R . [The factor 3/2 appears be¬ 
cause the gain of the target relative to an isotropic 
radiator is (3/2) Hence 


4 wffiWr = £ G r P s . 

z 


(52) 


(53) 


(54) 


Eliminating P s from equations (49) and (52), 

Wr = 9 \ 2 (V 
IF,- 16t r 2 d 2 ’ 

Putting this value of W r /Wi in equation (41), 

• - ?«»’. 

4tt 

which is the required general formula connecting 
target gain and radar cross section. It will be noted 
that the factor 9 X 2 / 47 r is just the radar cross section 
of the shorted doublet. 

Inserting the value of <r given by equation (54) 
into equation (45), 


Pi 


= 4GiGtf B * 


(—V 

\8 irdJ 


(55) 


which is the radar gain formula for free space in 
terms of the gain of the target relative to a dummy 
doublet. 

The reasonableness of the factor 4 in the above 
equation may be made apparent by the following 
analogy. Compare the doublet antenna with a 
generator whose internal resistance corresponds 
to the radiation resistance of the antenna. When the 
generator is shorted all the power is dissipated in 
the internal resistance. When the doublet is shorted 
all the power is reradiated. The maximum power 
that can be extracted from either the generator or 
the antenna occurs when the load resistance equals 
the internal generator, or antenna radiation, re¬ 
sistance. It is 34 the above short-circuit power. 
This is the 4 that occurs in the above equation. 

Equation (55), in the nonfree-space case, takes the 
form 

— = 4ff 1 G^G B t (—Y 
Pi \8 ird) 

where A P is the path gain factor defined by equation 

( 20 ). 


A 4 

A-p , 


(56) 



Chapter 3 
ANTENNAS 


31 FUNDAMENTALS 

31,1 Function of Antennas 

A transmitting antenna converts the power 
delivered to it into electromagnetic radiation 
(neglecting losses); a receiving antenna abstracts 
power from an incident electromagnetic wave and 
delivers to the receiver that part which is not re¬ 
radiated or lost in the antenna. In the short and 
microwave region the power conversion is effected 
with a very small loss so that for most practical 
purposes the power loss inside the antenna may be 
disregarded. Apparent losses caused by reflection 
owing to mismatch between the antenna and its 
input circuit are of a different nature and are not 
included herein. 

For many purposes it is desirable to concentrate 
the power radiated into a beam of comparatively 
small angle as in this way the field strength in the 
preferred direction is enhanced. The gain of a 
directional antenna is defined by means of a com¬ 
parison of the given antenna radiation pattern with 
that of an electric doublet. 

The gain of an antenna is the ratio of power that 
must be supplied to a doublet to the power that must 
be supplied to the antenna considered in order that, 
at a given large distance, the electric field at the 
maximum of the antenna pattern is equal to the 
field at the same distance in the equatorial plane of 
the doublet. From the reciprocity principle it is 
found that the gain of a receiving antenna is equal 
to the gain of the same antenna used as a transmitter. 
A discussion of antenna gain and reciprocity is given 
in Chapter 2, Section 2.1. 

312 Directive Antennas 

Polar plots of antenna radiation patterns are of 
two kinds: either the relative magnitude of the 
Poynting vector (power per unit area) is plotted 
along the radius vector, or the relative magnitude of 
the radiation electric field strength is plotted in the 
same way. Usually the value of the radius vector at 


the maximum of the pattern is taken equal to unity. 
The Poynting vector plot is obtained from the field 
strength plot by squaring the radial distances 
(Figure 1). 



If an antenna system is designed so that most of its 
power is concentrated into a comparatively small 
cone, the corresponding part of the radiation pattern 
is called the main lobe. Commonly there are a num¬ 
ber of secondary maxima (side lobes) much smaller 
than the main lobe. The vidth of the main lobe is 
measured by the angle between half-power points. 
Half-power points are those points in the polar 
diagram of the antenna pattern where the power 
per unit area is equal to one-half that at the maxi¬ 
mum, the field strength being l/V 2 = 0.707 times 
that at the maximum. This angle is also referred 
to as the beam width. The beam width varies from a 
degree or less for some specialized radar antennas to 
very large angles such as 50 to 60 degrees, depending 
on the design and purpose of the antenna. The 
larger the beam width the smaller the gain. 

It should be noted that an antenna radiation 
pattern may have high directivity with respect to 
one plane going through the antenna and little or 
no directivity in another plane. Thus a doublet 
antenna (for definition see Section 2.1.1) is directive 
in a plane which contains the antenna itself but is 
nondirective in the equatorial plane perpendicular 
to the antenna (see Figure 11). 


22 




FUNDAMENTALS 


23 


3,1,3 Antenna Pattern Factors 
in Ground Reflection 

With highly directive antennas the magnitude 
of the direct wave may differ appreciably from that 
of the ground-reflected wave owing to their differ¬ 
ence in angle of emergence from the antenna (Fig¬ 
ure 2). This must be taken into account by using the 


fiav 


Figure 2. Antenna pattern factors. 

antenna pattern factors Fi and F 2 in computing the 
interference pattern above the line of sight. This 
subject is dealt with in Section 5.2.6. 



dipole, shortened the right amount; or (2) the an¬ 
tenna system is made resonant by adding suitable 
reactive components to the radiative elements. To 
illustrate, the center-fed half-wave dipole of exactly 
half-wavelength, assuming sine distribution of cur¬ 
rent, has an inductive reactance; it may be made 
resonant by the addition in series of a capacitive 
reactance. This is known as antenna loading and is 
common at the longer wavelengths where half-wave 
dipoles would be too cumbersome. Another example 
is that of a dipole radiator shorter than the half¬ 
wave dipole and having the form of a metallic tube; 
this is combined with a tunable cavity resonator 


•-LESS THAN RESONANT LENGTH -* 


l 

i 


1 

i 



*! 

t i 

TUBE IN PUT X 

* 

/ 

inner'coaxial line 

-METAL SUPPORT 


Standing-Wave Antennas 


Figure 3. Antenna tuned to resonance by a shunt 
impedance. 


An important class of antennas is that in which 
standing waves of the currents and the voltages are 
set up. In a transmitting antenna of this type, for 
instance, a progressive or traveling wave is supplied 
from the connected source of power. This is re¬ 
flected from the end of the antenna and the inter¬ 
action of the two sets of waves moving in opposite 
directions results in a standing-wave system. 

In this event the current amplitude is zero at the 
ends of the antenna and assumes differing values 
at the other positions on the antenna. The dis¬ 
tribution of current amplitudes is usually assumed 
to vary sinusoidally with the distance from the end 
of the antenna. This is a good approximation where 
the diameter of the antenna wire is small compared 
with the length, but may be seriously in error for 
thick-wire antennas. 

The simplest, and one of the most commonly 
used, standing-wave antennas is the half-wave 
dipole antenna, discussed in Section 3.2.3. 

3,1,5 Resonant Antennas 

Many antennas are operated at or near resonance, 
which means that the reactive component of their 
impedance vanishes or is very small. 

Two types of resonant antenna may be dis¬ 
tinguished: either (1) the radiating element as a 
whole is resonant, as in the case of the half-wave 


inside the tube that acts as a shunt impedance, the 
whole system being tuned to resonance (Figure 3). 

Although the actual antenna impedance is made 
up in a complicated way of distributed capacitances 
and inductances, the input impedance of the simpler 
types of antennas for a limited frequency band 
containing the resonance frequency is essentially 
that of an ordinary series resonant circuit [the resist¬ 
ance at resonance being essentially the radiation 
resistance of the antenna (se6 Section 3.1.7)]. The 
input impedance of certain other antennas is essen¬ 
tially that of parallel-resonant circuits with very 
large shunt resistances at resonance (see Section 
3.2.2). For illustration, see Figure 6. 

3,1,6 Traveling-Wave Antennas 

In this type of antenna there is no standing-w r ave 
system set up since the progressive or traveling 
wave of current fed into the antenna is absorbed, 
without reflection, by a terminal resistance placed 
at the end of the antenna, which is equal to the 
characteristic impedance of the antenna regarded 
as a transmission line. Such antennas are necessarily 
nonresonant. 

The traveling-wave antenna radiates most strongly 
in the general direction of the w r ave motion. The 
major lobe makes an angle a < 90 degrees with this 

















24 


ANTENNAS 


direction as indicated in Figure 20. Here we have a 
long-wire antenna with the input at the left and the 
characteristic impedance (resistance) at the right. 

A traveling-wave V antenna uses two of these 
elements (see Section 3.3.2) and a rhombic is com¬ 
posed of four elements (see Section 3.3.3), with the 
elements arranged at angles which produce maximum 
directivity of the combinations. 

Antennas of the nonresonant or traveling-wave 
types are used both for longer and for very short 
waves. (However, there is an intermediate fre¬ 
quency region extending from about 100 to 3,000 me 
where the half-wave dipole is of such convenient 
size that standing-wave dipoles or dipole arrays are 
most frequently employed.) 

In the microwave band where transmission is 
effected by wave guides it is possible to terminate a 
wave guide with a horn which “matches the imped¬ 
ance of the wave guide to that of free space” and 
acts as a directive antenna (Section 3.7). A slot 
or a series of slots in the side of a wave guide may 
also act as an antenna at these frequencies. 


Radiation Resistance 


The radiation resistance R r of an antenna is the 
ratio P, of the total power radiated in all directions 
to the square of the current at the point of measure¬ 
ment. The power may be computed by integrating 
the radial component of the Poynting vector over a 
spherical surface surrounding the antenna. Then 
if Ii is the effective value of the input current, 


The radiation resistance of the doublet antenna is 
stated in equation (9) in Chapter 2 to be 



ohms. 


(2) 


3,1,8 Influence of Near-by 

Conducting Bodies 

The impedance of an antenna is affected by the 
presence of conductors in the vicinity and depends 
upon the mutual impedances between the conductors 
and the antenna. The mutual impedance decreases 
with increasing distance so that for conducting 
bodies of comparable size the effect is negligible for 


distances greater than, perhaps, 2 to 3 wavelengths. 

But for conductors set less than a wavelength 
apart, such as an antenna and reflector (or director) 
combination or as antenna arrays, the mutual 
effect plays an important role and modifies the 
input impedance of the antenna. 

For an antenna set near a large conducting body, 
such as a large metallic sheet or the earth, the mutual 
effect is cared for in a different way. If the earth, for 
instance, is assumed plane and perfectly conducting, 
its effect is the same as that of the mirror image 
of the antenna in the ground. As shown in Figure 4, 


T 

h 

h 

± 


It 


i ANTENNA 


i 



’///// /////7 / 


||i IMAGE 

I 1 


PLANE EARTH 

////V777 7 ///V '////ft/ ' */ 

PERFECT CONDUCTOR 


VERTICAL HORIZONTAL 


Figure 4. Method of images. 


the image of a vertical antenna is a similar antenna 
with current in the same direction, while the current 
is reversed for a horizontal antenna. The radiation 
field at any point above ground is obtained by 
summing the radiation fields of antenna and image. 


3 2 STANDING-WAVE ANTENNAS 

3,2,1 Linear Antennas 

A linear antenna is a straight thin rod supplied 
with alternating current. According to whether the 
connection to the antenna is made at the middle or 



END FED 

ALTERNATE CURRENTS 

Figure 5. Distribution 
linear antennas. 



CENTER FED 
CO-PHASED CURRENTS 

current amplitudes with 














STANDING-WAVE ANTENNAS 


25 


at the end, center-fed and end-fed antennas are 
distinguished. Center-fed linear antennas are also 
called dipole antennas. 

Typical current amplitude distributions are illus¬ 
trated in Figure 5. The amplitude is always zero 
at the open end while the amount at the input point 
depends on the position of the input connection. For 
thin wires, compared with the length, the distribu¬ 
tion of amplitudes is approximately sinusoidal. 


3,2,2 Half-Wave Antennas 

Figure 6 illustrates two types of half-wave dipole 
or center-fed antennas and one end-fed antenna, 
together with their lumped-circuit analogues. The 



CURRENT-FED OR 
CENTER-FED DIPOLE 


ACTUAL CIRCUIT 



END FED 


SCHEMATIC CIRCUIT 

e i=£=r- c=i=r 

A B C 


LUMPED-CIRCUIT ANALOGUE 



SERIES RESONANCE PARALLEL RESONANCE 


Figure 6. Three methods of exciting half-wave an¬ 
tennas and their analogues in lumped-constant resonant 
circuits. 


input current required varies with the position of 
the input point. The voltage distribution in general 
has a maximum at the points of current zero and 
has a minimum where the current is maximum. 


3,2,3 Half-Wave Dipole 

The half-wave dipole, shown in A and B of 
Figure 6 and in Figure 7, is the type most frequently 
used in the 100 to 3,000 me range. In this range the 
length X/2 lies between 1.5 and 0.05 meters. In this 
section it is assumed that the current distribution 
is sinusoidal. 


1. Radiation field. The radiation field at point P, 
Figure 7, where d > > X, is obtained by dividing 
the half-wave current distribution into an infinite 


P 



number of infinitesimal doublets, using equation (2) 
in Chapter 2 and taking into account the differences 
in phase at P introduced by the differences in the 
distances which the radiation from the various 
doublets must travel. The net result, using d in 
place of r, is 

E, = Mi voltspermeter, (3) 

d sin 9 

E 

H\ — —— amperes per meter. (4) 


The normal part of the field, E e (difference), pre¬ 
scribes the antenna pattern factor (measured in 
relative field strength) and is plotted in Figure 11. 
The corresponding pattern for a doublet is E e ~ sin 9, 
which is a circle in polar coordinates. These patterns 
are circularly symmetric about the antenna axis. 
Squaring the radial lengths in the above patterns 
gives the pattern in terms of relative power per unit 
area in the same angular direction. 

The radial component of radiated power per square 
meter (Poynting’s vector) is given by 



watts per square meter. 


In the equatorial plane, 


E e 


60 Ij 

d 


( 6 ) 




































26 


ANTENNAS 


2. Gain of half-wave dipole. The gain of the dipole 
relative to a doublet is the ratio of the power supplied 
to the doublet to the power supplied to the dipole to 
produce the same field strength at the same distance 
in the direction of maximum radiation (here the 
equatorial plane, 6 = 90 degrees). 

For equal maximum fields, comparing equations 
(3) in Chapter 2 and (6) in this chapter, 

f Idl = -1,. (7) 

J 7r 


The power per unit area for the doublet, using 
equation (3), in Chapter 2, is 


W 


doublet 


E 2 SOIi 2 sin 2 


120 7T 


t rd 2 


( 8 ) 


and for the dipole the power per unit area is given 
by equation (5). 

The dipole gain is then 

J JFdoubiet dA p ower ra 0iated by doublet 

/ Power radiated by dipole ’ 
fbdipole dA 

where the integration is carried out over spheres 
surrounding the antennas. Carrying out this opera¬ 
tion, 

Cdipoie = 1-09 (or 0.4 db). (9) 

3. Radiation Resistance. The radiation resistance 
of the half-wave dipole is 

Rr = ~ J ^Fdipoie dA = 73.1 ohms. (10) 

4. Impedance of an Infinitely Thin Dipole, f The 
formulas given here are valid only for a half-wave 



Figure 8. Half-wave dipole field components. 

dipole composed of wire of vanishing thickness. 
For wire of finite dimensions, see Section 3.2.7. 


Here (type A in Figure 6) it is necessary to calcu¬ 
late the voltage Vi required at the input to establish 
a current distribution I { cos [(2x/X)«], as shown in 
Figure 8. To do this, the total field of the dipole 
must be known, including the induction field which is 
significantly large at short distances as well as the 
radiation field. In cylindrical coordinates, the total 
field is given by 



By the reciprocity theorem a small current length 
I z dz = Ii cos [(2w/\)z] • dz induces a voltage (— dV { ) 
at the input point which is equal to the voltage 
dV z = E z dz induced in dz b}^ a small current length 
Iidz taken at the input point. Hence 


E z dz — dV\ 



and the total input voltage is 



Carrying out the operation indicated and dividing 
by Ii gives the impedance of the half-wave dipole as 

Z = 73.1 + j42.5 ohms. (14) 

The dipole thus has an inductive reactance of 42.5 
ohms if a sine distribution of current amplitudes is 
assumed. 

The reactance can be altered by changing the 
length of the wire. Increasing the length increases 
the inductance; decreasing the length decreases the 
inductance, first to zero for resonance, and then 
for still shorter lengths to a capacitive reactance. 
Changes in length of only 4 to 5 per cent will pro¬ 
duce large changes in the reactance. 


3,2,4 Modifications of the 

Half-Wave Dipole 

Two modifications will be given. 

1. Quarter-wave dipole with artificial ground. A 
convenient device for doubling the effective length 

















STAN DING-WAVE ANTENNAS 


27 


of a dipole is to use an artificial ground plane. It 
usually takes the form of a number of grounded 
rods spreading radially from the base of the antenna 



Figure 9. Quarter-wave dipole with artificial ground. 

(Figure 9). If the antenna is a quarter-wave dipole 
the effect of the artificial ground is to produce an 
image quarter-wave dipole; the radiation resistance 
and the radiation pattern of the system are those of a 
half-wave dipole. 



Figure 10. Folded dipole. 


2. Folded dipole. Another variant of the dipole 
antenna is the folded dipole, shown in Figure 10. 


It is essentially a center-fed half-wave dipole with a 
parasitic counterpart “dummy” (see Section 3.5) 
in its immediate neighborhood and connected to the 
latter at the ends of the dipole. The induced current 
in the dummy has the same distribution as, and is in 
phase with, that of the primary dipole. Hence the 
radiation pattern is essentially that of a simple half¬ 
wave dipole. The radiation resistance is four times 
that of the ordinary dipole. 

3 2 5 Multiple Half-Wave 

Long Antennas 

For an antenna of length equal to an integral 
number, n, of half wavelengths, the radiation field is 
given by: 

1. n is odd: 



2. n is even: 



where d is the radial distance to a field point and 
li is the input current at the center of one of the 
half-wave elements. 

The radiation patterns are illustrated in Figure 11 
for the doublet, n — 1 (the half-wave dipole), and 
n = 2, 3, 4. 



Figure 11. Antenna radiation patterns (relative field strength). 


n=4 























28 


ANTENNAS 


The radiation resistance, both for integral and 
nonintegral numbers of half wavelengths, is plotted 
in Figure 12. 



Figure 12. Radiation resistance for linear antennas. 

In Table 1 the radiation resistances and the power 
gains for integral half-wavelength antennas are 
listed. 


3 2 6 Cophased Half-Wave Dipoles 

The directivity and gain of linear antennas may be 
increased considerably by the suppression of alter¬ 
nate current loops, leaving therefore only loops in 
which the currents are all cophased. The suppressed 
loops are contained in either (1) quarter-wave stubs 
or (2) short inductive elements, as indicated in 
Figure 13. The suppressed loops are practically 
nonradiative. 

The radiation field at distance d is the vector sum 
of the fields from the n half-wave elements. The 
contribution from each element lags that of the next 
element above by an angle 

a = — cos 6 • — = 7r cos 0 radians, (17) 
2 X 


Table 1. Comparison of alternate and cophased half-wave dipoles. 


n 

Half 

waves 

Rn 

Radiation 

resistance-ohms 

Em,n 

Relative major lobe amplitudes 
for same current input 

Gn 

Gain (power) 

Alternate 

currents 

Cophased 

currents 

Alternate 

currents 

Cophased 

currents 

Alternate 

currents 

Cophased 

currents 

1 

73.1 

73.1 

1.0 

1 

1.09 

1.09 

2 

93 

199 

1.23 

2 

1.19 

1.47 

3 

105 

317 

1.38 

3 

1.32 

2.09 

4 

113 

439 

1.5 

4 

1.46 

2.67 

5 

121 

560 

1.62 

5 

1.58 

3.26 


^doublet /^doubletV _ ^doublet / Em < n \ 2 
R~ n \ I n ) ~ R„ \ ^doublet/ 



Figure 13. Cophased half-wave dipoles. 































































































STANDING-WAVE ANTENNAS 


29 


determined by the extra distance [(X/2) cos 0] 
which it must travel. The radiation field is then 
equal to the radiation field of one half-wave element 
(as a function of angle 0) multiplied by the vector 


The radiation patterns for various values of n are 
plotted in Figure 14. Table 1 gives the radiation 
resistances, relative lengths of major lobes, and the 
gains, with comparative figures for the doublet and 





Figure - 14. Cophased half-wave dipoles (relative fields). 


resultant for the n elements. Thus 



[e j0 + e* + e> 2a + + + e j{n ~ 1)a ] 


60 1, 

COS 

\J COS0 ) 


. na~ 
Sln ~2 

d 


sin 0 


a 

L sin T J 


(18) 


the multi-half-wave antennas of Section 3.2.5. 


3,27 Effects of Finite Diameter 
on Center-Fed Linear Antennas 

Figure 15 shows the input reactance, and Figure 16 
the input resistance of a center-fed antenna of 
arbitrary length. The input impedance is a series 
combination of the two components. The important 



HALF-LENGTH OF ANTENNA 

Figure 15. Reactance at input of a center-fed antenna of arbitrary length. 













































30 


ANTENNAS 


regions of the curves correspond to antenna half- 
lengths near X/4 and near A/2. The former repre¬ 
sents a center-fed half-wave antenna, whereas the 
latter represents a pair of end-fed half-wave antennas 
excited in phase. The half-length of the antenna was 
used in plotting, because in these terms the reactance 
curves resemble those for an open-ended trans¬ 
mission line. 

In the regions of principal interest the reactance 
curves are nearly straight lines whose slopes depend 
on the diameter of the antennas expressed in wave¬ 
lengths. The slopes of the reactance curves decrease 
as the antenna diameter increases. This feature is 
important in radar antennas which need to be in¬ 
sensitive to small changes in frequency. The curves 


than when the half-length approximates A/4, as it 
does for a single center-fed antenna. The values for 
an antenna whose half-length is X/4 is not readable 
on the curve, but the component representing radia¬ 
tion ranges from 73 ohms for infinitely thin antennas, 
through 64 ohms for a diameter of 0.0001 X, 55 ohms 
for a diameter of 0.01 X, to less than 50 ohms for 
certain large-diameter radar antennas. The change 
is mainly due to a decrease in the resonant length of 
the thicker antennas. 

A feature of Figure 15 which is not easily readable 
is that the lengths at which the reactance is zero are 
less than X/2 and X. The amount by which an an¬ 
tenna with zero reactance is shorter than these 
lengths depends on the antenna diameter. For very 



HALF-LENGTH OF ANTENNA 


Figure 16. Resistance at input of a center-fed antenna of arbitrary length. 


show that antennas of large diameter present less 
than a specified amount of reactance, say one ohm, 
over a greater range of antenna length than slender 
antennas do. In terms of frequency, this means 
that a given length of antenna has less than one-ohm 
reactance over a wider range of frequency when the 
antenna has a large diameter than when it has a 
small diameter. Radar antennas are commonly 
made of tubing and frequently have diameters in 
excess of A/20. 

Figure 16 shows that the input resistance also 
depends on antenna diameter. This dependence is 
more pronounced when the half-length is about X/2 


slender antennas the shortening is slight, but for 
large-diameter antennas or for special shapes as 
shown in Figure 17, a resonant length may be as 



Figure 17. Non-cylindrical half-wave antenna. 

much as 20 per cent shorter than X/2. Special shapes, 
such as the one shown in Figure 17, have the ad- 

























STANDING-WAVE ANTENNAS 


31 


'vantage of being insensitive to small changes in 
frequency and at the same time are not so subject to 
corona (breakdown of the air because of large poten¬ 
tial gradients) as slender antennas are. 

328 Standing-Wave V Antennas 

This type of antenna (Figure 18) utilizes the 
directive properties of the multi-half-wave antenna. 


Two such elements are combined in a V arrange¬ 
ment so that the major lobe of each (at angle a with 
each element) is parallel to the axis of the V. By 
feeding the two halves of the V with currents 
180 degrees out of phase the lobe structure is reversed 
to produce maxima, forward and backward, along 
the axial direction, while the field in the plane 
perpendicular to the axis is greatly reduced. The 


A 



Figure 18. Standing-wave V antenna (« = 36° for n — 4 half-wavelengths). 



IN PLANE OF THE V 

.O 


IN PLANE -L TO V 

O 


0 = 16 , of = 17.5“ n ' 16 > a - l7 - 5 

Figure 19. Power distribution for standing-wave V antenna. (Courtesy of IRE ) 











































32 


ANTENNAS 


value for angle a is equal to the angle between each 
■element and its maximum lobe (see Figure 11). 
Figure 19 gives the (power) radiation pattern for 
n = 16 half-wavelengths. 

The directivity of this antenna system may be 
improved by adding one or more reflectors (see 
Section 3.4.6). The reflector is a V antenna of 
identical type. The legs of the reflector are placed 
parallel to those of the primary V and lie in the same 
plane as the original V. The reflector is set approxi¬ 
mately X/4 behind the primary V. 

33 TRAVELING-WAVE ANTENNAS 

331 Field and Pattern 

A traveling-wave antenna is one in which only 
progressive (or traveling) waves are allowed. Re¬ 
flected waves are eliminated by terminating the end 
opposite the input point in the characteristic imped¬ 
ance. See Figure 20. 


The major lobes given by this equation are plotted 
in Figure 21, and the major lobe angles with the 
wire 6 m are plotted in Figure 22. Angle 6 m , it will be 
noted, decreases with increasing wire length. 

3 3 2 Traveling-Wave V Antenna 

As in the case of the standing-wave antenna a 
pair of lines arranged at a suitable angle with each 



Figure 20. Lobe structure for L = 2\ traveling-wave 
antenna in free space. 



Figure 21. Major lobes (relative field strength) for traveling-wave antenna. 


The equation of the radiation field, neglecting 
wire losses, is 



other, and carrying traveling waves, can be made to 
produce a directional pattern with fairly high gain. 

The traveling-wave V antenna can be designed 
so that the plane of the V is horizontal and the 
maximum lies in the direction of the axis of sym- 










ANTENNA ARRAYS 


33 


me try, as in Figure 18. In this case the radiation is 
horizontally polarized. It can also be used as an 
inverted V in a vertical plane with the point of the 
V directed upwards; the radiation is then vertically 
polarized. This antenna, also called a semi-rhombic, 
is represented by the upper half of Figure 20. 



L/X 


Figure 22. Angles for major lobes for traveling-wave 
antenna. 


The field in the axial direction is equal to 
„ 2407; cos 0 . [ ttL , . .1 . v 

E axial = —— - T^wSin 2 — (1 - sin 0) . (20) 

a 1 - sin p LX J 

Effect of Perfectly Conducting Ground. If the 
rhombic is placed in a horizontal plane, height H 
above ground, the effect of the image rhombic 
must also be considered. The net result is that the 
direction of the resultant lobe maximum is tilted 
up by an angle e. It can be shown that, for a given 
angle e and wavelength X, to point the major lobe 
at vertical angle e the following relations for 77, 
L, and 0 must hold: 



4 sin e ’ 


L _ 0.371X 

sin 2 e ’ 

0 = 90° - e . 

Figure 24 illustrates the radiation pattern (relative 
field strength) for a particular case. 


Rhombic Antenna 


This type of antenna is based on the same prin¬ 
ciple as the traveling-wave V antenna. The rhombic 
antenna consists of four wires arranged in the form 
of a rhomboid or diamond (Figure 23). The reflec¬ 
tionless termination of the wires is achieved by 
connecting the two wires at the end opposite the 
input to a resistance equal to their characteristic 
impedance. 



Figure 23. Rhombic antenna. 


As in the case of the V antenna, the rhombic 
antenna can be used both horizontally and vertically; 
at the longer waves the horizontal arrangement is 
usually more practical. The optimum tilt angle of 
the rhombic (angle 0 of Figure 23) is not very 
critical provided the legs are not less than two wave¬ 
lengths long. The radiation pattern is not very 
sensitive to frequency and the rhombic antenna can 
therefore be used over a fairly wide frequency range 
(of the order of 2 to 1). Rhombic antennas have 
appreciably higher gains than V antennas. 


ANGLE IN DEGREES 


10 


0 PLAN VIEW 


L * 4.1 X 
H s 0.83 X 

t -17.5° 


20 

€ - 17.5° 

ELEVATION 

10 

0 

2 3 

Figure 24. Rhombic antenna above ground (relative 
field strength). (Courtesy of Bell System Technical 
Journal.) 

34 ANTENNA ARRAYS 

3,4,1 Principle of Arrays 

An antenna array is a combination of several 
antennas, usually of equal strength and equally 
spaced in any one given direction. One-, two-, and 
three-dimensional arrays may be distinguished. The 









































34 


ANTENNAS 


spacings in different directions may be different for 
two- or three-dimensional arrays. The use of arrays 
permits great increases in the amount of power 
radiated, in direct!vitjq and gain. 

Although the most common array element is a 
half-wave dipole, the elements of an array may be 
radiators of any type; in particular, the elements 
may themselves be arrays. In this way it is possible 
to interpret a two-dimensional array as an array of 
arrays. A vertical curtain may be considered either 
as a horizontal array of elements which, themselves, 
are vertical, or it may be considered a vertical array 
of elements which, themselves, are hoiizontal arrays; 
similarly for three-dimensional arrays. 

In most arrays the elements radiate very nearly 
equal power, but in the binomial array the elements, 
although identical in structure, differ in the amount 
of power radiated because of differing current dis¬ 
tributions. In most arrays there is a constant phase 
shift (which might be zero) between adjacent ele¬ 
ments. By suitable phasing a great variety of an¬ 
tenna patterns can be produced. 

3,4 2 Basic Types of Dipole Arrays 

There are three basic types of dipole arrays. 

1. Broadside array. The centers of the elements 
are arranged in a line, with the axes of the elements 
parallel to each and perpendicular to the line. With 
the currents adjusted all in phase, the maximum 
radiation is broadside to the plane of the elements. 

2. End-fire array. The geometric arrangement is 
the same as in the broadside array, but through 
appropriate phasing of the currents in the elements 
the maximum radiation can be directed primarily 
along the line joining the centers. 

3. Colinear array. Here the axes of the antenna 
elements are arranged along the line of centers with 
the currents all in phase. The radiation is a max¬ 
imum in the equatorial plane perpendicular to the 
line of centers. 

To illustrate the principles most simply, two half¬ 
wave dipole elements are considered first, and later 
extension is made to arrays composed of a larger 
number of elements. 

3 4 3 Two-Dipole Side-by-Side Array 

Tw r o half-wave dipoles are placed side by side with 
spacing s and the currents Ii and I 2 are equal but 
differ in phase by angle \f/ (see Figure 25). If J 2 lags 


1 1 by time angle \j/, the field of the second element 
at P lags that of the first by angle a where a is 



composed of \f/ and the time delay caused by the 
extra distance traveled, (2tt/\)s cos </> sin 0, 

a = \J/ + (27r/X) s cos </> sin 0. (21) 

For equal currents, /1 = 1 2 = /, the field is equal 
to 


E* 


60 /Lo + ^I 

d [_ J [_ sin 0 


60/ 

sin a 


cos cos dj 

d 

. a 

L 8ln 2 J 


sin 0 


( 22 ) 


The first bracket gives the directional characteristic 
of an array of two elements, while the second bracket 
gives jthe directional characteristic of the i element 
itself. 



BROADSIDE END-FIRE UNI-DIRECTIONAL 

3 s X/2 S=X/2 S*X/4 

f--0° ^ = 180° ^ = 90°< I 2 LAGS I, ) 

EQUATORIAL PLANE 0=90° 

Figure 26. Radiation patterns (field strength) for two 
dipole side-by-side array. 


Three special cases are particularly to be noted. 
The field patterns for the equatorial plane (0 = 90°) 
and | I 2 | = | Ii | are plotted in Figure 26. 
















ANTENNA ARRAYS 


35 


1. Broadside. Here $ = X/2, the currents are in 
phase (\f/ = 0°). The maximum field is broadside and 
twice that of each dipole. 



Figure 27. Two half-wave dipole colinear array. 


3 4 4 Two-Dipole 

Colinear Array 

For two equal currents in time phase (see Figure 
27), the field is equal to 



The field is circularly symmetrical about the axis. 
Its variation with 0 is plotted in Figure 28. This is, 
of course, equivalent to a vertical antenna with 
center height at distance s/2 above a perfectly 
conducting flat earth. 






Figure 28. Two half-wave dipole colinear array. 


2. End-fire. Again s = X/2, but the currents are 
out of phase = 180°). The maximum field is 
found in both directions along the line of centers. 

3. Unidirectional couplet. Here s = X/4 and J 2 
lags 1 1 by \p = 90°. The result of this combination 
is to produce a maximum field along the line of 
centers in the direction looking from the leading to 
the lagging current and zero field in the reverse 
direction. 


One-Dimensional Array 

Two geometrical arrangements will be considered. 
1. Broadside. Consider n elements with equal co¬ 
phased currents equally spaced (see Figure 29). 

. na 
sin — 

E 0 = E e i (0, <t>) -, (25) 

a 

sm T 


























36 


ANTENNAS 



Figure 29. Broadside array, elements perpendicular 
to paper. 



n = 5 


rt = 9 


n=i3 


n = i7 


Figure 30A. Effect of array length for broadside with 
element spacing s = X/2. (From Radio Engineers’ 
Handbook by Terman.) 



Figure 30B. Effect of element spacing for broadside 
for array length L = 3X. (From Radio Engineers’ 
Handbook by Terman.) 


where 


2tt 

a — - s COS g>. 

X 


(26) 


For center-fed half-wave dipoles, from equation (3) 


E, = 


607 


cos 


(i cos e ) 


sin 6 


(27) 


The patterns for the equatorial plane are illustrated 
in Figures 30A and 30B. Figure 30A shows the in¬ 
crease in directivity with increasing number of 
elements. Figure 30B illustrates the effect of element 
spacing on the production of side lobes. Figure 31 
gives the gain for various spacings and array lengths. 
From this it appears that s = 5X/8 is approximately 
the optimum spacing. 



Figure 31. Gain for broadside array of doublets as a 
function of array length and spacing. (From Radio 
Engineers’ Handbook by Terman.) 





















































































ANTENNA ARRAYS 


37 


2. Broadside: pattern factor and beam width. For 
illustration, suppose that the antenna consists of a 
vertical array of m horizontal center-fed dipoles 
spaced s apart with all fed in phase to give a broad¬ 
side beam strongly directive in the vertical plane. 
For this arrangement the field strength in the 
horizontal plane is given by m times equation (27), 
that is, 


E 


horizontal 


m 

d 


cos 




(28) 


sin 9 


with angle 9 measured from the dipole axis. See also 
equation (15) and Figure 11 (for n = 1). 

In the vertical plane, the beam is much narrower. 
If cj) is the angle from the vertical and /3 = 90° — 
is the angle from the (horizontal) broadside direc- 
t ion, the field in the vertical plane (9 = 90°) is given 

i>y 


E 


vertical 


with 


. ma 
sin — 
= 607,_ 2_ 

d . a 
sin - 
2 


27t ,27r . 

a = — s cos 9 = — s sin (3. 


(29) 


The maximum value of equation (29), corresponding 
to 9 = 90° and 0 = 0°, is equal to 
„ 607; 

E max = ——m 

d 

The relative field strength is then 

ma 


^vertical 2 


E r 


m sin 


(30) 


( 31 ) 



Figure 32. Cophased colinear half-wave dipoles. 

The beam width in the vertical plane is determined 
by the angle between the half-power points, or the 
angle between the points where the field strength is 


2c 


JC 


CM 


t: 


A 



-i 

/ Line of 

- ! 

i- -j 

/ Array 



u 

i 



/< (ft 

CO - 


_l C 


Midpoint Spacing s= X/2 
Effect of Array Length 



*< I CM 


CD 



Array Length = 3X 
Effect of Element Spacing 

Figure 33. Cophased colinear half-wave dipoles. (From Radio Engineers’ Handbook by Termano) 






































38 


ANTENNAS 


0.707 of the maximum. Thus, equation (31) is set 
equal to 0.707 and 0 determined. The beam width is 
equal to 2/3°. 

To illustrate, let $ = X/2. For m = 2,3,4,8,12,16 
dipoles in the array, the corresponding beam widths 
are 60°, 36.4°, 26.4°, 12.8°, 8.5° and 6.3°. A few field 
patterns are illustrated in Figure 30A and gains are 
shown in Figure 31. 

3. Colinear array (see Figure 32). For n equal 
cophased currents, equally spaced, 

. na 
sin — 

E e = E el {6,4>) -(32) 

. a 
sm 2 

where 

a — — s cos 6 . (33) 

X 

For center-fed half-wave dipoles, from equation (3), 



The patterns for various array lengths and spacings 
are given in Figure 33. 

If $ = X/2, equation (32) for half-wave dipoles 
reduces to equation (18). 


34 8 Binomial Arrays 

Most array patterns show, in addition to the main 
maximum, secondary maxima (side lobes) which are 
inconvenient in radar work. Side lobes are practi¬ 
cally eliminated in the binomial array. 


Q*n/2 



B -1*1- 


(B) Colinear 


Unidirectional Broadside 
and Colinear Arrays 

If an array is backed up with a similar array, the 
latter may serve to concentrate the radiation in one 
direction, provided the currents in the arrays are 
properly adjusted in magnitude and phase. Patterns 
for the unidirectional broadside and colinear arrays 
are given in Figure 34. 

The broadside array is a collection of unidirec¬ 
tional couplets of the type illustrated in Figure 26. 
Increasing the number of couplets n appreciably 
narrows the beam width. 

A similar improvement is obtained in the colinear 
combination. 

Multidimensional Arrays 

Enough has already been given to show that it is 
not difficult to extend the principles of summation 
of fields from elements to cover two- and three- 
dimensional arrays. 


Figure 34. Unidirectional broadside and colinear array. 

Consider first a two-element half-wave dipole 
broadside array with equal cophased currents and 
elements spaced a half wavelength apart. See equa¬ 
tions (21) and (22). Then^ = 0 ,s = X/2, a = r cos<£. 
Then with 



E = E el («*> + e~n. 

This gives the broadside field in Figure 26 which has 
only two equal major lobes. There are no side lobes. 
Now r consider the equation 

E = E el (e j0 + e~ ja y , (36) 

= E el (1 + 2e~ ia + le~ j2a ). 

This represents three half-wave dipoles in broadside, 
spacing s = X/2, currents in phase but w ith relative 
magnitudes 1 : 2 : 1. The pattern has no side lobes. 









PARASITIC REFLECTORS AND DIRECTORS 


39 


Again, for five similar dipoles, 

E = E d (e*° + O 4 

= E d { 1 + 4e- Ja + 6e~ j2a + 4e” j3 “ + le~ j4a ). K J 

Here the cophased current magnitudes are 1:4: 
6 : 4 :1, and there are no side lobes. 

The scheme is to follow the pattern of binomial 
coefficients in adjusting the relative current values. 
m — 1 11 

2 12 1 

3 13 3 1 

4 1 4 6 4 1 

5 1 5 10 10 5 1 

Each number is the sum of the two immediately 
above. 

34 9 Ring Arrays 

A set of radiating elements can be arranged on the 
perimeter of a circle with equal angular distances and 
equal phase shift between the elements; the diameter 
of the ring must be properly chosen; the resulting 
radiation pattern can be made very nearly uniform, 
i.e., circular, in the plane of the ring, while the 



directivity of the pattern obtained in a plane per¬ 
pendicular to that of the ring is increased compared 
to that of the single element. 

If a number of such rings are stacked on top of each 
other with a common vertical axis, a linear array is 
formed whose elements are the rings. A radiation 
pattern is thus produced that has pronounced direc¬ 


tivity in elevation while it is nearly uniform in 
azimuth. This device is frequently used in micro- 
wave beacons. 

The most common form of the ring antenna is the 
triple dipole shown in Figure 35. The elements are 
three half-wave dipoles spaced 120 degrees apart. 

3 5 PARASITIC REFLECTORS 

AND DIRECTORS 

3 51 Parasitic Antennas 

A parasitic antenna or ‘‘dummy” is an antenna 
which is not connected to the antenna input termi¬ 
nals; if placed in the vicinity of a driven antenna k 
current is induced in the former which modifies the 
radiation field. Parasitic antennas provide a simple 
means of producing a moderate increase of directiv¬ 
ity. Depending on the relative phase of the currents 
in the two antennas, the maximum of the radiation 
pattern either is found in the direction of the parasite 
and the latter is then called a director; or it is found 
in the direction of the primary element and the para¬ 
site is then called a reflector. In order to obtain good 
directive action the two dipoles must be close to¬ 
gether, that is, a fraction of a wavelength. 

352 Half-Wave Dipole and Parasite 

The geometrical arrangement corresponding to 
the following discussion is illustrated in Figure 36. 
The radiation field at any point in the equatorial 
plane (6 = 90°) is equal to 

E„ = ( ^h e > a -|- 60/i e yo- ( 2»A).co.*] ) 
d d 

where Jo is the center-fed input current to the an¬ 
tenna, 7 1 is the center value of the current in the 
parasite, and 0 is the angle by which 7i leads J 0 . 
The relation between 7 1 and 7 0 is given by 

h = 09) 

I ^11 I 

where Z 0 1 is the vector mutual impedance of antenna 
and parasite, and Z n , the vector self-impedance of 
the parasitic antenna, is equal to Rn + jXu = 

| Zn | e js . 8 is the phase angle of the parasite self¬ 
impedance. 











40 


ANTENNAS 


The field pattern in^the equatorial plane is de¬ 
pendent directly on the spacing and indirectly also, 
since the spacing controls the mutual impedance and 
thus the voltage induced in*the parasite. The current 


ohms; if longer, the inductive reactance is increased; 
if shorter, it becomes at first less inductive, then 
resonant with Xu = 0, and finally, capacitatively 
reactive. Field patterns for s/X = 0.1 and s/X = 0.25, 


Power 





Power 


Figure 36. Half-wave dipole and parasite. 


in the parasite is further dependent on its self¬ 
impedance, which can be changed by altering the 
length of the parasite. Cut to a length of just X/2, 
the self-impedance is inductive, Z n = 73.1 + J42.5 



Figure 37. Relative field of half-wave antenna and 
parasite. (Courtesy of I. R. E.) 


and for 8 = +22.5°, 0°, and —22.5°, are plotted 
in Figure 37. These illustrate that, by controlling 
the spacing and length of parasite, it is possible to 
direct the pattern maximum into either the R or D 
direction, so that the parasite acts primarily either 
as a reflector ( E R > E D ) or as a director (E D > E R ). 

1. Parasite as a reflector. For good reflector per¬ 
formance, the spacing s/X should lie between 0.15 
and 0.25 with the parasitic element made slightly 
longer (perhaps 5 per cent) than X/2 in order to 
increase its inductive reactance. A few of the 
equatorial field patterns are shown in the lower row 
of Figure 37. To obtain the strongest field in the 
R direction, it is necessary to lengthen the parasite 
to a particular length (obtained by trial). If this is 
done, Figure 38 indicates that the field E R is a 
maximum for s/X = 0.15 and that the ratio of 
E r to E for the antenna alone is 1.83; for s/X = 0.25 
it is 1.65. This does not, however, give the best 
front-to-back ratio. 





































PARASITIC REFLECTORS AND DIRECTORS 


41 


2. Parasite as a director. Good director perform¬ 
ance is obtained when s/\ = 0.1 and the parasitic 
element is cut slightly shorter (perhaps 4 per cent) 
than X/2 to produce a capacitative reactance. See 
Figure 37, upper row, for the field patterns and 



ER/Eantenna alone 
Ed/E antenna alone 


Reflector 

Director 


Figure 38. Adjustment of parasite for strongest 
fields Er and E D . (Courtesy of I. R. E.) 


Figure 38 for the best ratio of E D to E for the antenna 
alone. The latter, again, does not give the best 
front-to-back ratio. 


Multiple Parasites. 
Yagi Antennas 


The most commonly used of the multiparasitic 
arrays of half-wave dipoles is the Yagi antenna 
(Figure 40). It has one reflector and several (usually 
2 to 5) directors. Since the voltage at the center of a 
dipole is always zero, it is possible to weld all the 



Figure 40. Yagi antenna with three directors. 

parasites to a central sustaining rod, as shown. By 
increasing the number of directors, it is possible to 
/ obtain highly directive patterns. 

The spacings between the elements of a Yagi 
array are not uniform. They are determined so that 
the phase difference of the currents in adjacent ele¬ 
ments is equal to their distance expressed in wave¬ 
lengths. If this condition is fulfilled, the elements 
are in phase with respect to radiation in the D 
direction. In practice, the spacing is determined 
experimentally rather than by calculations, which 
become very cumbersome when several directors 
are employed. 

3,5,4 Reflecting Screens 


By using several parasites, rather pronounced 
directive effects can be achieved. Figure 39 shows a 
typical example. This antenna uses three parasitic 


FIELD 

3 PARASITES PATTERN 


a =.248 X 
b=.588X 
C=.535X 


'o b 

c — c • 

t 

DRIVEN 

ELEMENT 



Figure 39. Antenna with three parasite elements. 
(From Radio Engineers’ Handbook by Terman.) 


dipoles arranged in a triangle or parabolic curtain. 
In order to obtain the most favorable pattern in 
such cases, careful tuning of the parasites is required. 


A plane-conducting screen placed behind a radiat¬ 
ing dipole has a similar effect in the forward direction 
as an image dipole whose distance s from the primary 
dipole is twice that of the screen and which has a 
phase shift of 180° from the primary dipole. Radia¬ 
tion in the backward direction is confined to the 
weak fields leaking around the edges of the screen. 
The pattern in the forward direction is given by the 
array formula of equation (22), and end-fire array 
with = 180° and s = X/2. Good results are 
achieved when the distance from the screen to the 
dipole is small (less than X/4) but larger spacings 
are also used. The change in input impedance of the 
primary element caused by the presence of the screen 
is appreciable. 

Reflecting screens are used primarily in connection 
with broadside arrays (curtains) to eliminate one 





























42 


ANTENNAS 


of the two main lobes in opposite directions. An 
adequate screening effect is produced by a set of 
wires parallel to the direction of the radiating dipole 
with spacings somewhat less than a tenth of a 
wavelength. 


3 - 5 - 5 Corner-Reflector Antenna 

i 

A simple directional device that gives an appre¬ 
ciable power gain (of the order of 10 to 20) is a 
corner reflector, which is essentially a combination 
of two reflecting sheets and a dipole. In the case 
shown in Figure 41 where the angle subtended by 



the corner is 90°, the corner is equivalent to the 
combined radiation of three image antennas. The 
reflector can also be made of wires parallel to the 
direction of the radiating dipole. The reflecting 
wflres do not, however, act as parasitic antennas 
but are taken so long that they are practically 
equivalent to conductors of infinite length. 

These should not be confused wdth corner re¬ 
flectors which are extensively used as targets and 
consist then of three mutually perpendicular con¬ 
ducting planes (see Section 9.2.4). 


3 6 PARABOLIC ELEMENTS 

361 Parabolic Reflectors 

These reflectors are the devices most commonly 
used to produce highly directive radiation patterns 
in the microw r ave region. The three main types are 
shown in Figure 42; they are the parabolic cylinder, 
the paraboloid of revolution, and the truncated 
paraboloid, the latter being a rectangular section 
cut from a paraboloid of revolution. If the parabolic 


cylinder is relatively short and provided with flat 
metallic covers at the top and bottom, its shape and 




C TRUNCATED 
PARABOLOID 


Figure 42. Types of parabolic reflectors. 


its electrical properties resemble those of a sectoral 
horn (see Section 3.7.2). 

The directive action of the parabolic reflector 
depends on two geometrical properties of the parab- 



Figure 43. Properties of a parabola. 


ola (Figure 43). A ray coming from the focus is 
reflected into a direction parallel to the axis of the 
parabola, and the distance from any point P on the 
parabola to the line called the directrix is equal to 
the distance from P to the focus. Consequently, the 
effect of the parabola in the forward direction is 
equivalent to that of a distribution of sources in the 
directrix that all oscillate in phase (but usually have 
varying intensities over the directrix). 

The parabolic cylinder produces a directive pattern 
only in a plane perpendicular to the generating line 
of the cylinder (horizontal plane in A of Figure 42). 
In order to concentrate the beam in a plane parallel 
to the generating line of the cylinder (vertical plane 
in Figure 42), an additional directive device must be 
employed. Usually this is a colinear array of dipoles, 
as shown; the direction of polarization is parallel 
to the focal axis. In microw r ave work this type of 
antenna offers advantages over the tw r o-dimensional 
curtain of dipoles employed in VHF directional 
antennas. 














PARABOLIC ELEMENTS 


43 


For the paraboloid of revolution or the truncated 
paraboloid, a simple source of radiation at the focus 
is used. Often this is a half-wave dipole, sometimes 
combined with a parasitic dipole which acts as a 
reflector (Section 3.5.2). 

In other types, the energy is brought to the focal 
point by a wave guide and is then reflected back onto 
the parabolic surface. 

If the wavelength is small compared with the 
dimensions of the parabolic reflector, the following 
approximate formula holds for the radiation pattern 
produced by a parabola: 


E = constant 



r tD . 01 

sin 

— sin -- 

L 2 X J 


ttD sin 0/X 


(1 + cos 0), 


(40) 


where D is the aperture of the reflector and 0 the 
angle from the axis. The half-power points corre¬ 
spond approximately to 

sin d = d = —— . (41) 

D 

These formulas correspond to the case of nearly 
uniform illumination of the reflector from the source 
at the focus. In practice a source that concentrates 
the field toward the center of the parabola is used 
in order to reduce the magnitude of the side lobes. 




Figure 45. Radiation pattern for a sectoral horn hav¬ 
ing various flare angles. 


The half-power angle is then more nearly equal to 
0 = 0.6X/D. 

The maximum gain of a parabolic reflector is 

°-(t )';■ <“> 

For D = 2 meters and X = 0.1 meter, the gain is 
approximately 1,000. 




Horizontal Aperture in X 


Figure 46. Gain of sectoral horn with TEi,o wave. 
(These curves are for a vertical aperture ratio a /X = 1. 
For other ratios the gain given should be multiplied by 
a /X.) (From Radio Engineers’ Handbook by Terman.) 
















































































44 


ANTENNAS 


37 HORNS 

3,7,1 Types of Horns 

Many of the antennas previous^ described are 
used in the high-frequency [HF] and very high- 
frequency [VHF] bands of frequencies. Horns cannot 
readily be used at these frequencies because the sizes 
required would be excessive. 

But at the microwave frequencies, the size of the 
horn is small and it is easy to feed energy to it 
through a wave guide. In this arrangement the 
horn acts as transition between the impedance of 
the wave guide and the 377 ohms impedance of 
free space and thus reduces to a minimum the 
reflection of energy backward into the guide (such 
as would occur if the wave guide ended in an open 
pipe). 

Common types of horns are sectoral (discussed 
in Section 3.7.2), pyramidal, conical, biconical, etc. 
Only the first type is discussed in this section. 


3,7,2 Sectoral Horn with TE h0 Wave 


For this case the horn is flared only in width and 
is an extension of the wave guide of width b and 
depth a. For the TE l 0 wave the electric field is 
parallel to the dimension a and varies in strength 
cosinusoidally across the wave guide and horn open¬ 
ing, as shown in Figure 44. The length of the horn 
is R and the flare angle is <t>. 

Figure 45 illustrates the pattern shapes in the 
plane parallel to dimension b for various flare angles. 

For this type of wave, the cutoff frequency of the 
wave guide is 


_ 3 X 1 0 8 
c 2b 9 


(43) 


with b in meters. The operating frequency should 
be near but not greater than twice this value. 

The gain depends upon the length R and the flare 
angle </>, and is plotted in Figure 46. 




Chapter 4 

FACTORS INFLUENCING TRANSMISSION 


41 REFRACTION 

411 Survey 

R efraction is caused by the variation of the 
. dielectric constant (square of refractive index) 
of the atmosphere. Although the atmosphere is 
tenuous and the variations of refractive index 
are small, the effect of refraction upon the field- 
strength distribution of waves is considerable. As 
will be shown, refraction under average conditions 
may be taken into account by using an earth with a 
modified radius. A representative average value of 
modified earth radius commonly used is ka with 
k = 4/3. Under certain conditions, especially in 
warmer climates, a slightly higher value of k might 
be preferable. 

The case where a is replaced by 4a/3 is referred to 
as standard refraction. It corresponds to a linear 
variation of refractive index with height in the 
atmosphere. In recent years, more complicated 
variations of refractive index in the atmosphere 
have received considerable attention and have 
proved to be of great operational interest. This 
volume, however, is restricted to consideration of 
standard atmosphere propagation. 

412 Snell’s Law 

Let n 0 and denote the refractive indices of two 
media separated by a plane boundary. The ordinary 
law of refraction known as Snell’s law is then usually 
stated (see Figure 1), as 

n 0 sin 0 O = Wi sin 0i, 

where 0o and 0i are the angles which the ray makes 
with the perpendicular to the boundary. It is con¬ 
venient to use the complementary angle a, so that 

n Q cos a:o = Wi cos a\. 

For several plane-parallel boundaries, Snell’s law 
generalizes to 

n 0 cos a 0 = ni cos ai = n 2 cos a 2 = • • • . 

In the atmosphere, the refractive index is a con¬ 
tinuous function of the height. Again, it is usually 


legitimate to consider the atmosphere as horizon¬ 
tally’ stratified, so that the refractive index is a 
function of height only. The case of a continuously 
variable refractive index is readily obtained by 



Figure 1. Refraction at boundary between two 
media. 


passing to the limit of an infinity of parallel bound¬ 
aries infinitely close together, Snell’s law remaining 
the same; thus 

n(h) • cos a = n Q cos a 0 , 

where now n and a are continuous functions of the 
height. In place of a discontinuous change in direc¬ 
tion, there will now occur a bending of the rays 
(Figure 2). 



Figure 2. Refraction in the atmosphere with variable 
n(h). 


If the boundaries are not plane but spherical, 
Snell’s law must be modified. Analysis shows that 
over a spherical earth surrounded by an atmosphere 
in which the refractive index n is a function of the 
distance r from the earth’s center, the law of re¬ 
fraction becomes 

n(r) • r cos a = n 0 r 0 cos ao , (1) 

where a is the angle between a ray and the horizontal 
(see Figure 3). 


45 





40 


FACTORS INFLUENCING TRANSMISSION 


Refraction is of practical importance only when 
the angle between the rays and the horizontal is 
small. In the determination of gain as given in later 



Figure 3. Refraction over curved earth. 


n — n 0 = constant X h. Then equation (2) may be 
written in the form 

“ (« 2 “ <*o 2 ) = (5) 

2 ka 

where k is the factor mentioned in Section 4.1.1 
which determines the modified earth’s radius ka. 
Comparing the above expression with equation (2), 
and differentiating, it follows that 


or 


_ 1 _ - _|_ I 

ka dh a 


k = 


1 + a 


dn 

dh 


1 

a 



( 6 ) 


chapters, the effect of refraction becomes com¬ 
pletely negligible when a is more than a few degrees. 

For small angles, cos a may be replaced by 
1 — a 2 / 2. In this case equation (1) is well approx¬ 
imated by 

— (a 2 — ocq 2 ) = n — Uq -|—, (2) 

2 a 


Proof of the fact that refraction is negligible unless 
the angle is very small may readily be deduced from 
the preceding formulas. Thus, on differentiating 
equation (4), 

10 " 6 

da = dM . —, 
a 

and in the standard linear case, by equation (5), 


where h is the height above the ground, so that 
r 0 = a and r = a + h. This is the practical form of 
Snell’s law for the atmosphere above a curved earth. 
The reference level (see Figure 3) is here taken at 
the surface of the earth where n Q is the index of 
refraction. 

4 *\ 3 X Modified Refractive Index 

In place of the sum (n 0 + h/a ) that appears in 
equation (2), it is customary to define and use a 
quantity M given by 



M is called the modified refractive index. It gives a 
unit that is convenient for practical use. The modi¬ 
fied index is then said to be expressed in M units, 
values of which commonly lie in the range of 300 to 
500. Using this definition, equation (2) becomes 

- (a 2 - a„ 2 ) = (M — Mo). 10- 6 . (4) 

2 

An important special case is that in which the 
refractive index decreases linearly with height, 


da = dh/kaa ~ 1.2 • 10 7 dh/a 

for k = 4/3. Taking a = 0.05 radians (3°) and 
dh = 100 meters, one finds da = 0.00024 radians 
(50 seconds of arc), a very small change in angle. 
This is the standard deflection which is accounted 
for by replacing a by ka. The deviations from this 
value experienced with nonstandard refraction are 
even smaller. The larger the angle a with the hori¬ 
zontal at which a ray issues from the transmitter, the 
less the angular deviation. In communication work 
and for certain radar problems, however, angles of 
less than one degree are of importance, and da 
may then become comparable to a. 

4.\.4 Graphical Representation 

Figures 4 to 6 show three different ways of repre¬ 
senting rays subject to refraction. Figure 4 gives a 
true picture apart from the exaggeration of heights. 
In the case of standard refraction, the curvature 
of the rays is always concave downwards, the center 
of curvature being below the surface of the earth. 
The middle ray shown is the horizon ray and to the 
lower right is the diffraction region into which rays 
do not penetrate. Figure 5 shows a diagram with 









REFRACTION 


47 


modified earth’s radius, ka, in which the rays are 
straight lines. Figure 6, finally, is a plane earth 
diagram; the rays are here curved upwards. 



Figure 4. Ray curvature over earth with radius a. 



Figure 5. Rays in a homogeneous atmosphere 
(equivalent radius ka). 

These diagrams may be considered as resulting 
from each other by changing the earth’s curvature 
by an arbitrary factor. From this viewpoint Figure 6 



Figure 6. Rays in a plane earth diagram. (Radius 
of curvature of rays is — ka.) 

h’l 


I 



Figure 7. Equivalent parabolic earth diagram. 

represents the limiting case of an infinite earth’s 
radius. The plane earth diagram is widely used for 
problems of nonstandard propagation. 


In drawing diagrams for a curved earth of equiva¬ 
lent radius ka, it is customary to replace the spher¬ 
ical earth outline by an equivalent parabola (see 
Figure 7). The equation for the surface reduces from 
the circular form, 

z , 2 + (h s + kaY = (kaY, 

to the parabolic form, 



for h s << x s . The height h measured from the 
surface of the earth, instead of from the x axis, is 
given by 


in which h is laid off perpendicular to the x axis and 
not to the earth’s surface. For clarity in drawing 
rays or field-strength diagrams, the vertical scale 
is expanded by an arbitrary factor p, whence 



This distortion of vertical distances, it can be shown, 
does not distort angles. The parabolic representa¬ 
tion to be reasonably accurate must be restricted to 
heights in the atmosphere small compared with the 
extent of the horizontal scale. 

4,1,5 Curvature Relationships 

The curvature of a ray is defined as the reciprocal 
of the radius of curvature p. Let \f/ be the angle 
between the ray and a nearly horizontal x axis. 



Figure 8. Angular relationships of rays. 

By Figure 8, p = — ds/d\J/, and since ^ is a small 
angle we may, to a sufficient approximation, put 
ds = dx, so that 

1 = _ # 

P dx 



















48 


FACTORS INFLUENCING TRANSMISSION 


Here the curvature has been defined so that it is 
positive when the ray curves in the same direction 
as the earth; vith this system the curvature of the 
earth itself is positive. Referring to Figure 8B, 

p dx dx dx 

But dcj)/dx = 1/a, and since a is a small angle 

da _ da dh _ da _ 1 d{a 2 ) 

dx dh dx dh 2 dh 


Consequently, by equation (2) 

1 _ _ 1 d(a 2 ) _|_ J_ _ dn 

p 2 dh a dh 


(9) 


From this, the curvature of the ray is equal to the 
vertical rate of decrease of the refractive index. 
Notice that dn/dh is usually negative, so that the 
true curvature of a ray is usually concave downwards. 

A simple relationship exists between m = p/a, 
the ratio of the radius of curvature of a ray to the 
radius of the earth, and k. Combining equations (6) 
and (9) gives 


\ + -= !• 


( 10 ) 


Consider again the special case where dn/dh = 
constant, so that n is a linear function of the height 
(standard refraction). Consider the plane earth 
diagram of Figure 6. The angles between corre¬ 
sponding curves are the same as in the true diagram, 
Figure 4. Hence, for the plane earth diagram, 
equation (8) becomes 1/ p' = — da/dx, where p' is 
the radius of curvature of the ray in the plane earth 
representation. It is readily found that 



dh ka 


Since M usually increases with height, the curvature 
of rays is concave upwards in this diagram. Again, 
equation (11) shows that when the modified earth’s 
radius ka is introduced (Figure 5), this amount of 
upward curvature is just canceled and the rays 
appear as straight lines. 


416 Alternate Method 

Instead of taking account of refraction by chang¬ 
ing the earth’s curvature, another method is some¬ 
times more convenient. It may be shown that the 


ratio of the field E to the free-space field E 0 trans¬ 
forms in the same way, whether (1) radius a is 
replaced by ka, or (2) the horizontal distance x is 
replaced by xk~ 2/ 3 and at the same time all elevations 
h are replaced b yhk~ 1/3 . An angle a must then be 
replaced by ak~ 1/s . Method (2) is usually less con¬ 
venient than method (1) because it involves a change 
of horizontal distance which makes it necessary to 
transform the ratio E/E 0 rather than the field itself. 
In method (1) where only curvatures are changed, 
this difficulty does not appear as the distance x 
and hence E 0 remains unaltered. 

Method (2) may be used to advantage to account 
for deviations of k from the standard value of 
k = 4/3. Coverage diagrams are usually drawn 
for this value; the deviations owing to a change in 
k may then be estimated by multiplying distances, 
heights, or angles with the appropriate powers of 

*/( 4/3). 

417 Computation of Refractive Index 

The following equation gives the dependence of 
the refractive index on temperature, pressure, and 
humidity: 

(n-l).10 6 = p(p + ^| 0 -) (12) 

where T is the absolute temperature, p is the total 
pressure, and e the water-vapor pressure, both the 
latter in millibars. 

Introducing M from equation (3), the modified 
refractive index, for use on a plane earth diagram, 
is equal to 

M = ™(p + ^ S ) + 0.157h, (13) 

where the height h is in meters. If h is in feet, the 
last term is 0.048 h. 

Tables have been prepared by means of which M 
can be computed rapidly from meteorological data, 
namely temperature, humidity, and pressure given 
as a function of height. For this purpose M is the 
sum of three terms which are computed independ¬ 
ently: 

M = M d -J- M w + M c . (14) 

The dry term M d is obtained from Table 1 as a 
function of temperature and height in meters above 
the ground. (If the pressure at the ground p 0 is 
substantially different from 1,000 millibars all 
values of M d should be multiplied by p 0 /l,000. In 




REFRACTION 


49 


t(° C) 
hi m)\ -20 


-18 


-16 


-14 


-12 


Table 1 

-10 


-8 


-6 


-4 


-2 


±0 


Hit) 


0 

312.3 

309.8 

307.4 

305.0 

302 7 

300.4 

298.1 

295.9 

293.7 

291.5 

289.4 

0.0 

10 

311.9 

309.4 

307.0 

304.6 

302.3 

300.0 

297.7 

295.5 

293.3 

291.1 

289.0 

32.8 

20 

311.5 

309.0 

306.6 

304.2 

301.9 

299.6 

297.3 

295.1 

292.9 

290.8 

288.7 

65.6 

30 

311.0 

308.6 

306.2 

303.8 

301.5 

299.2 

297.0 

294.8 

292.6 

290.4 

288.3 

98.4 

40 

310.6 

308.2 

305.8 

303.4 

301.1 

298.8 

296.6 

294.4 

292.2 

290.1 

288.0 

131.2 

50 

310.2 

307.8 

305.4 

303.0 

300.7 

298.4 

296.2 

294.0 

291.8 

289.7 

287.6 

164.0 

75 

309.2 

306.8 

304.4 

302.1 

299.8 

297.5 

295.2 

293.0 

290.8 

288.7 

286.6 

248.1 

100 

308.1 

305.7 

303.4 

301.1 

298.8 

296.5 

294.3 

292.2 

290.0 

287.9 

285.8 

328.1 

150 

306.0 

303.6 

301.3 

299.0 

296.8 

294.6 

292.4 

290.3 

288.2 

286.1 

284.0 

492.1 

200 

303.9 

301.6 

299.3 

297.1 

294.9 

292.7 

290.5 

288.4 

286.3 

284.2 

282.2 

656.2 

250 

301.9 

299.6 

297.3 

295.1 

293.0 

290.8 

288.7 

286.6 

284.5 

282.5 

280.5 

820.2 

300 

299.9 

297.7 

295.4 

293.2 

291.1 

288.9 

286.8 

284.7 

282.7 

280.7 

278.7 

984.3 

350 

297.8 

295.6 

293.4 

291.2 

289.1 

287.0 

285.0 

282.9 

280.9 

279.0 

277.0 

1,148.0 

400 

295.8 

293.6 

291.5 

289.4 

287.3 

285.2 

283.2 

281.1 

279.1 

277.2 

275.3 

1.312.0 

450 

293.8 

291.7 

289.6 

287.5 

285.4 

283.3 

281.3 

279.3 

277.3 

275.4 

273.5 

1,476.0 

500 

291.9 

289.8 

287.7 

285.6 

283.5 

281.5 

279.5 

277.6 

275.6 

273.7 

271.8 

1,640.0 

600 

288.0 

285.9 

283.8 

281.8 

279.9 

277.9 

276.0 

274.1 

272.2 

270.3 

268.5 

1,969.0 

700 

284.1 

282.1 

280.1 

278.1 

276.2 

274.3 

272.4 

270.5 

268.7 

266.9 

265.1 

2,297.0 

800 

280.3 

278.3 

276.4 

274.5 

272.6 

270.7 

268.9 

267.1 

265.3 

263.6 

261.8 

2,625.0 

900 

276.5 

274.6 

272.7 

270.9 

269.0 

267.2 

265.4 

263.7 

262.0 

260.3 

258.6 

2,953.0 

1,000 

272.8 

271.0 

269.1 

267.3 

265.6 

263.8 

262.1 

260.4 

258.7 

257.0 

255.3 

3,281.0 

1,500 

255.0 

253.4 

251.8 

250.2 

248.7 

247.2 

245.7 

244.2 

242.8 

241.3 

239.9 

4,921.0 

2,000 

238.3 

237.0 

235.7 

234.3 

233.0 

231.7 

230.4 

229.1 

227.8 

226.5 

225.3 

6,562.0 


-4.00 

-0.40 

+3.20 

6.80 

10.4 

14.C0 

17.6 

21.2 

24.8 

28.4 

32.0 

\h(ity 


t(° F) 



Table 1 ( Continued) 
M. 


a 


10 


12 


14 


16 


18 


20 


hi ft) 


0 

289.4 

287.3 

285.2 

283.2 

281.1 

279.2 

277.2 

275.3 

273.4 

271.5 

269.6 

0.0 

10 

289.0 

286.9 

284.8 

282.8 

280.8 

278.9 

276.9 

274.9 

273.1 

271.2 

269.3 

32.8 

20 

288.7 

286.6 

284.5 

282.5 

280.5 

278.5 

276.6 

274.6 

272.8 

270.9 

269.0 

65.6 

30 

288.3 

286.2 

284.1 

282.1 

280.1 

278.2 

276.2 

274.3 

272.4 

270.5 

268.7 

98.4 

40 

288.0 

285.9 

283.8 

281.8 

279.8 

277.8 

275.9 

274.0 

272.1 

270.2 

268.4 

131.2 

50 

287.6 

285.5 

283.4 

281.4 

279.5 

277.5 

275.6 

273.7 

271.8 

269.9 

268.1 

164.0 

75 

286.6 

284.7 

282.6 

280.6 

278.7 

276.7 

274.8 

272.8 

271.0 

269.1 

267.3 

248.1 

100 

285.8 

283.8 

281.7 

279.7 

277.8 

275.8 

273.9 

272.0 

270.2 

268.3 

266.5 

328.1 

150 

284.0 

282.0 

280.0 

278.0 

276.1 

274.2 

272.3 

270.4 

268.6 

266.8 

265.0 

492.1 

200 

282.2 

280.2 

278.3 

276.3 

274.4 

272.5 

270.7 

268.8 

267.0 

265.2 

263.4 

656.2 

250 

280.5 

278.5 

276.6 

274.7 

. 272.8 

270.9 

269.1 

267.2 

265.4 

263.7 

261.9 

820.2 

300 

278.7 

276.8 

274.9 

273.0 

271.1 

269.2 

267.4 

265.6 

263.8 

262.1 

260.3 

984.3 

350 

277.0 

275.1 

273.2 

271.3 

269.4 

267.6 

265.8 

264.0 

262.2 

260.5 

258.8 

1,148.0 

400 

275.3 

273.4 

271.5 

269.6 

267.8 

266.0 

264.2 

262.5 

260.7 

259.0 

257.3 

1,312.0 

450 

273.5 

271.7 

269.8 

268.0 

266.2 

264.4 

262.6 

260.9 

259.2 

257.5 

255.8 

1,476.0 

500 

271.8 

270.0 

268.1 

266.3 

264.6 

262.8 

261.1 

259.4 

257.7 

256.0 

254.4 

1,640.0 

600 

268.5 

266.7 

264.9 

263.1 

261.4 

259.6 

258.0 

256.3 

254.7 

253.0 

251.4 

1,969.0 

700 

265.1 

263.4 

261.7 

259.9 

258.2 

256.5 

254.9 

253.2 

251.6 

250.1 

248.5 

2,297.0 

800 

261.8 

260.1 

258.4 

256.7 

255.0 

253.4 

251.8 

250.3 

248.7 

247.2 

245.6 

2,625.0 

900 

258.6 

256.9 

255.2 

253.6 

252.0 

250.4 

248.8 

247.3 

245.8 

244.3 

242.8 

2,953.0 

1,000 

255.3 

253.7 

252.1 

250.5 

249.0 

247.4 

245.9 

244.4 

242.9 

241.4 

239.9 

3,281.0 

1,500 

239.9 

238.5 

237.0 

235.6 

234.3 

232.9 

231.6 

230.3 

228.9 

227.6 

226.3 

4,921.0 

2,000 

225.3 

224.1 

222.9 

221.7 

220.5 

219.3 

218.1 

217.0 

215.8 

214.7 

213.5 

6,562.0 


32.0 

35.6 

39.2 

42.8 

46.4 

50.0 

53.6 

57.2 

60.8 

64.4 

68.0 













tm \ 





50 


FACTORS INFLUENCING TRANSMISSION 


Table 1 ( Continued) 


H m)\ 

0 

10 

20 

30 

40 

50 

75 

100 

150 

200 

250 

300 

350 

400 

450 

500 

600 

700 

800 

900 

1,000 

1,500 

2,000 


tc C) 

20 

22 

24 

26 

28 

** 

30 

32 

34 

36 

38 

40 

/h(it) 

269.6 

267.8 

266.0 

264.2 

262.5 

260.7 

259.0 

257.5 

255.7 

254.0 

252.4 

0.0 

269.3 

267.5 

265.7 

263.9 

262.2 

260.4 

258.7 

257.2 

255.4 

253.7 

252.1 

32.8 

269.0 

267.2 

265.4 

263.6 

261.9 

260.1 

258.4 

256.9 

255.1 

253.4 

251.8 

65.6 

268.7 

266.9 

265.1 

263.3 

261.6 

259.9 

258.2 

256.5 

254.9 

253.2 

251.6 

98.4 

268.4 

266.6 

264.8 

263.0 

261.3 

259.6 

257.9 

256.2 

254.6 

252.9 

251.3 

131.2 

268.1 

266.3 

264.5 

262.7 

261.0 

259.3 

257.6 

255.9 

254.3 

252.6 

251.0 

164.0 

267.3 

265.5 

263.8 

262.0 

260.3 

258.6 

256.9 

254.2 

253.6 

251.9 

250.3 

248.1 

266.5 

264.7 

263.0 

261.2 

259.5 

257.8 

256.2 

254.5 

252.9 

251.3 

249.7 

328.1 

265.0 

263.2 

261.5 

259.7 

258.0 

256.3 

254.7 

253.1 

251.5 

249.0 

248.3 

492.1 

263.4 

261.7 

260.0 

258.3 

256.6 

254.9 

253.3 

251.7 

250.1 

248.5 

246.9 

656.2 

261.9 

260.2 

258.5 

256.8 

255.1 

253.5 

251.9 

250.3 

248.7 

247.2 

245.6 

820.2 

260.3 

258.6 

256.9 

255.3 

253.6 

252.0 

250.4 

248.9 

247.3 

245.8 

244.3 

984.3 

258.8 

257.2 

255.5 

253.9 

252.2 

250.6 

249.0 

247.5 

245.9 

244.4 

242.9 

1,148.0 

257.3 

255.7 

254.0 

252.4 

250.8 

249.2 

247.7 

246.2 

244.6 

243.1 

241.6 

1,312.0 

255.8 

254.2 

252.6 

251.0 

249.4 

247.8 

246.3 

244.8 

243.3 

241.8 

240.3 

1,476.0 

254.4 

252.8 

251.1 

249.5 

248.0 

246.4 

244.9 

243.4 

241.9 

240.4 

239.0 

1.640.0 

251.4 

249.8 

248.3 

246.7 

245.2 

243.7 

242.2 

240.7 

239.2 

237.8 

236.4 

1,969.0 

248.5 

247.0 

245.5 

243.9 

242.4 

240.9 

239.5 

238.0 

236.6 

235.2 

233.8 

2,297.0 

245.6 

244.1 

242.6 

241.2 

239.7 

238.3 

236.9 

235.4 

234.0 

232.7 

231.3 

2,625.0 

242.8 

241.3 

239.8 

238.4 

237.0 

235.6 

234.2 

232.8 

231.4 

230.1 

228.8 

2,953.0 

239.9 

238.5 

237.1 

235.7 

234.3 

232.9 

231.6 

230.3 

228.9 

227.6 

226.3 

3,281.0 

226.3 

225.1 

223.8 

222.6 

221.4 

220.2 

219.0 

217.8 

216.6 

215.5 

214.3 

4,921.0 

213.5 

212.4 

211.3 

210.3 

209.2 

208.1 

207.1 

206.0 

205.0 

203.9 

202.9 

6,562.0 

86.0 

89.6 

93.2 

96.8 

100.4 

104.0 

89.6 

93.2 

96.8 

100.4 

104.0 

t(° F) 

\ Hit) 


general this correction may safely be neglected un¬ 
less the difference between p 0 and 1,000 is quite 
large, corresponding to an elevation of the ground 
level of several thousand feet above sea level.) 

The wet term M w is obtained from Table 2 as a 
function of temperature and relative humidity. 
Finally, M c — 0.157 h, if h is in meters, or M c = 0.048 h, 
if h is in feet, may readily be computed by means of a 
slide rule. M is then obtained by addition. 

418 Atmospheric Stratification 

Ordinary weather data give comparatively little 
information about the atmospheric stratification 
near ground level. Radiosonde data are too widely 
spaced (vertical distances of the order of 100 meters 
between successive readings) for reliable determina¬ 
tion of the variation of M with height at low levels. 
Special instruments have therefore been developed 
in recent years for low-level soundings. Such instru¬ 
ments contain temperature and humidity measuring 
elements that are relatively free of lag; they are 
attached to airplanes or dirigibles or they are carried 
aloft by means of captive balloons or kites. The 
above tables are for use in connection with such 
measurements. 


Two main cases must be distinguished. First, the 
refractive index or M is very nearly a linear function 
of the height in the lower layers (at heights above 
about 500 to 800 meters the variation of refractive 
index with height will deviate from linearity only in 
very exceptional instances). This is the standard 
case where dM/dh is independent of h, and by using 
equation (6), k is conveniently obtained from the 
slope of the M — h curve. It is found that the 
vertical temperature gradient has a comparatively 
small influence on k, while fairly small variations 
of the humidity gradient will affect k appreciably. 

The other case is that of nonstandard refraction. 
Here M is not a linear function of the height. The 
most important special case is that of superrefrac¬ 
tion, which occurs when, in certain height intervals, 
M decreases with height instead of following the 
usual increase with elevation. Such a decrease of M 
in certain layers of the atmosphere is caused by a 
steep negative moisture gradient or steep positive 
temperature gradient, or even more by a combina¬ 
tion of both influences. 

With superrefraction, propagation conditions are 
greatly different from those encountered with 
standard refraction and the methods to be given 
later for the determination of the transmitted power 





REFRACTION 


51 


Table 2 
M 



10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

/ 

-20 

0.6 

1.2 

1.8 

2.5 

3.1 

3.7 

4.3 

4.9 

5.5 

6.130 

- 4.0 

-18 

0.7 

1.5 

2.2 

2.9 

3.7 

4.4 

5.1 

5.8 

6.6 

7.309 

- 0.4 

-16 

0.9 

1.7 

2.6 

3.5 

4.3 

5.2 

6.1 

6.9 

7.8 

8.678 

+ 3.2 

-14 

1.0 

2.1 

3.1 

4.1 

5.1 

6.2 

7.2 

8.2 

9.3 

10.287 

+ 6.8 

-12 

1.2 

2.4 

3.6 

4.8 

6.1 

7.3 

8.5 

9.7 

10.9 

12.124 

+10.4 

-10 

1.4 

2.9 

4.3 

5.7 

7.1 

8.6 

10.0 

11.4 

12.9 

14.284 

+ 14.0 

-8 

1.7 

3.4 

5.0 

6.7 

8.4 

10.1 

11.7 

13.4 

15.1 

16.754 

+17.6 

-6 

2.0 

3.9 

5.9 

7.8 

9.8 

11.7 

13.7 

15.7 

17.6 

19.568 

21.2 

-4 

2.3 

4.6 

6.9 

9.1 

11.4 

13.7 

16.0 

18.3 

20.6 

22.871 

24.8 

-2 

2.7 

5.3 

8.0 

10.6 

13.3 

16.0 

18.6 

21.3 

23.9 

26.589 

28.4 

±0 

3.1 

6.2 

9.3 

12.4 

15.5 

18.5 

21.6 

24.7 

27.8 

30.911 

32.0 

2 

3.5 

7.0 

10.5 

14.1 

17.6 

21.1 

24.6 

28.1 

31.6 

35.139 

35.6 

4 

4.0 

8.0 

12.0 

16.0 

20.0 

24.0 

28.0 

32.0 

35.9 

39.939 

39.2 

5 

4.3 

8.5 

12.8 

17.0 

21.3 

25.5 

29.8 

34.0 

38.3 

42.533 

41.0 

6 

4.5 

9.1 

13.6 

18.1 

22.6 

27.2 

31.7 

36.2 

40.8 

45.280 

42.8 

7 

4.8 

9.6 

14.5 

19.3 

24.1 

28.9 

33.7 

38.5 

43.3 

48.175 

44.6 

8 

5.1 

10.2 

15.4 

20.5 

25.6 

30.7 

35.9 

41.0 

46.1 

51.215 

46.4 

9 

5.4 

10.9 

16.3 

21.8 

27.2 

32.6 

38.1 

43.5 

49.0 

54.394 

48.2 

10 

5.8 

11.6 

17.3 

23.1 

28.9 

34.7 

40.5 

46.2 

52.0 

57.793 

50.0 

11 

6.1 

12.3 

18.4 

24.5 

30.6 

36.8 

42.9 

49.0 

55.1 

61.270 

51.8 

12 

6.5 

13.0 

19.5 

26.0 

32.5 

39.1 

45.6 

52.1 

58.6 

65.094 

53.6 

13 

6.9 

13.8 

20.7 

27.6 

34.5 

41.4 

48.3 

55.2 

62.1 

69.010 

55.4 

14 

7.3 

14.6 

22.0 

29.3 

36.6 

43.9 

51.2 

58.5 

65.9 

73.169 

57.2 

15 

7.8 

15.5 

23.3 

31.0 

38.8 

46.5 

54.3 

62.0 

69.8 

77.505 

59.0 

16 

8.2 

16.4 

24.6 

32.8 

41.0 

49.2 

57.4 

65.6 

73.9 

82.060 

60.8 


r.h. 


Table 2 ( Continued ) 
M 



t(° C)\ 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

/t(° F) 

16 

8.2 

16.4 

24.6 

32.8 

41.0 

49.2 

57.4 

65.6 

73.9 

82.060 

60.8 

17 

8.7 

17.4 

26.0 

34.7 

43.4 

52.1 

60.8 

69.5 

78.1 

86.813 

62.6 

18 

9.2 

18.4 

27.6 

36.7 

45.9 

55.1 

64.3 

73.5 

82.7 

91.866 

64 A 

19 

9.7 

19.4 

29.1 

38.9 

48.6 

58.3 

68.0 

77.7 

87.4 

97.127 

66.2 

20 

10.3 

20.5 

30.8 

41.1 

51.4 

61.6 

71.9 

82.2 

92.4 

102.71 

68.0 

21 

10.8 

21.7 

32.5 

43.4 

54.2 

65.1 

75.9 

86.8 

97.6 

108.46 

69.8 

22 

11.5 

22.9 

34.4 

45.8 

57.3 

68.7 

80.2 

91.6 

103.1 

114.56 

71.6 

23 

12.1 

24.2 

36.3 

48.3 

60.4 

72.5 

84.6 

96.7 

108.8 

120.87 

73.4 

24 

12.8 

25.5 

38.3 

51.0 

63.8 

76.5 

89.3 

102.0 

114.8 

127.53 

75.2 

25 

13.4 

26.9 

40.3 

53.8 

67.2 

80.7 

94.1 

107.6 

121.0 

134.44 

77.0 

26 

14.2 

28.3 

42.5 

56.7 

70.9 

85.0 

99.2 

113.4 

127.6 

141.74 

78.8 

27 

14.9 

29.9 

44.8 

59.8 

74.7 

89.6 

104.6 

119.5 

134.5 

149.39 

80.6 

28 

15.7 

31.5 

47.2 

62.9 

78.7 

94.4 

110.1 

125.9 

141.6 

157.34 

82.4 

29 

16.6 

33.1 

49.7 

66.2 

82.8 

99.4 

115.9 

132.5 

149.1 

165.62 

84.2 

30 

17.4 

34.9 

52.3 

69.7 

87.1 

104.6 

122.0 

139.4 

156.8 

174.27 

86.0 

31 

18.3 

36.7 

55.0 

73.3 

91.7 

110.0 

128.3 

146.7 

165.0 

183.32 

87.8 

32 

19.3 

38 5 

57.8 

77.1 

96.4 

115.6 

134.9 

154.2 

173.5 

192.74 

89.6 

33 

20.3 

40.5 

60.8 

81.0 

101.3 

121.5 

141.8 

162.0 

182.3 

202.56 

91.4 

34 

21.3 

42.6 

63.9 

85.1 

106.4 

127.7 

149.0 

170.3 

191.6 

212.86 

93.2 

35 

22.4 

44.7 

67.1 

89.4 

111.8 

134.1 

156.5 

178.8 

201.2 

223.50 

95.0 

36 

23.5 

46.9 

70.4 

93.9 

117.3 

140.8 

164.2 

187.7 

211.2 

234.63 

96.8 

37 

24.6 

49.2 

73.9 

98.5 

123.1 

147.7 

172.3 

197.0 

221.6 

246.20 

98.6 

38 

25.8 

51.6 

77.5 

103.3 

129.1 

154.9 

180.8 

206.6 

232.4 

258.23 

100.4 

39 

27.1 

54.2 

81.2 

108.3 

135.4 

162.5 

189.6 

216.6 

243.7 

270.80 

102.2 

40 

28.4 

56.8 

85.2 

113.6 

142.0 

170.4 

198.8 

227.2 

255.5 

283.94 

104.0 





52 


FACTORS INFLUENCING TRANSMISSION 


do not apply. This is especially true for the field 
near or below the optical horizon. The discussion of 
nonstandard propagation is beyond the scope of this 
volume. 

High-angle coverage is generally independent of 
refraction and is therefore also unaffected by the 
variations of M in the lowest layers of the atmosphere. 

Direct Determination of k 

In Figure 9 the reciprocal of k is plotted as ordinate 
against the vertical gradient of relative humidity as 
abscissa. The values shown refer to a standard 
temperature gradient of — 0.65 degrees Centigrade 
per 100 meters; unless the temperature gradient 
differs greatly from this value, the corresponding 
values of k will not be much affected. The curves 
given refer to 100 per cent relative humidity at 
ground level, and an auxiliary table is provided 


4 2 GROUND REFLECTION 

421 Ground Reflection and Coverage 

The reinforcement of the direct ray by the ground- 
reflected ray is of great importance both in radar 
and in communication work. In favorable cases, 
the reflected ray may be of comparable intensity 
to the direct ray and thus the received intensity 
may be approximately doubled in places where the 
two rays have the same phase. This means, in many 
cases, the possibility of an appreciable increase in 
range relative to that in free space. 

4 2 2 Complexity of Reflection Problem 

In order to facilitate the discussion of the problems 
encountered in reflection, it is necessary to analyze 
the complex phenomena into simpler constituents. 


1.2 



(RH)„ 

- 30 C 

- 25 C 

- 20 C 

- 15 c 

- 10 C 

- 5 C 

0 C + 5 C + 10 C + 15 C - 4 - 20 C + 25 C + 30 C + 35 C 4 - 40 C 

10 % 

.009 

.012 

‘.017 

.027 

.039 

.055 

.078 .090 

7118 

.151 

.194 

.240 

.301 

.371 

.455 

20 % 

.008 

.011 

.015 

.024 

.034 

.049 

.069 .080 

.105 

.134 

.172 

.214 

.267 

.329 

.405 

30 % 

.007 

.009 

.014 

.021 

.030 

.043 

.060 .070 

.092 

.118 

.151 

.187 

.234 

.288 

.354 

A 0 % 

.006 

.008 

.012 

.018 

.026 

.037 

.052 .060 

.079 

.101 

.129 

.161 

.200 

.247 

.304 

| 30 % 

.005 

.007 

.010 

.015 

.021 

.031 

.043 .050 

.066 

.084 

.108 

.134 

.167 

.206 

.253 

1 00 % 

.004 

.005 

.008 

.012 

.017 

.024 

.034 .040 

.052 

.067 

.080 

.107 

.134 

.165 

.202 

70 % 

.003 

.004 

.006 

.009 

.013 

.018 

.026 .030 

.039 

.050 

.065 

.080 

.100 

.123 

.152 

80 % 

.002 

.003 

.004 

.006 

.009 

.012 

.017 .020 

.026 

.034 

.043 

.065 

.067 

.082 

.101 

90 % 

.001 

.001 

.002 

.003 

.004 

.006 

.009 .010 

.013 

.017 

.021 

.027 

.033 

.041 

.051 


-5 -6 -7 -8 -9 


-10 -II -12 -13 


Figure 9. Graph 1/Jc versus RH (relative humidity) gradient and temperature for 100 per cent RH at ground. Add 
correction tabulated to obtain 1/k for RH at ground less than 100 per cent. 


at the top of the graph which gives figures to be 
added for other values of the relative humidity. 
The sensitivity of k to moisture gradients in warm 
•climates is obvious from these data. 


First of all, the incident radiation field is resolved 
into nearly plane wave components, each forming a 
narrow pencil of rays striking the reflecting surface 
within a small area which we shall call the reflection 
























































































GROUND REFLECTION 


53 


point. There are two types of such rays depending 
on their state of polarization. If the electric vector 
is parallel to the reflecting plane, the rays are said 
to be horizontally polarized and if the electric vector 
is parallel to a vertical plane through the rays, they 
are said to be vertically polarized. When consider¬ 
ing a very irregular surface, the reflected field may 
show extreme complexity even though the incident 
wave is linearly polarized. Increasing roughness 
may result in diffuse reflection which is ineffective 
in reinforcing the direct wave. The existence of 
diffuse reflection depends primarily on the size of the 
irregularities of the surface in comparison with the 
wavelength of the incident radiation and on the 
grazing angle of the incident field. This problem 
will be discussed in more detail later. 

4 2 3 Plane Reflecting Surface 

Consider first the simplest case, when a plane 
wave strikes a plane surface such as that of an 
absolutely calm sea. The incident ray is then split 
into two parts. One is the reflected ray, which is 
returned to the atmosphere, and the other is the 
refracted ray, which is absorbed by the sea. At the 
point of reflection, the ratio of any scalar quantity 
in the reflected wave to the same quantity in the 
incident wave is defined as the reflection coefficient 
of the sea for plane waves of given frequency. Thus 
defined, the reflection coefficient can and will be 
different for the various components of the field. 

For simplicity, let us assume the reflecting plane 
to be the xy plane of a rectangular coordinate system, 
the xz plane to coincide with the plane of incidence, 
and the reflection point to be the origin of the 
coordinate system. 



. PLANE OF INCIDENCE 




^^-INCIDENT RAY 

| REFLECTED RAY.. 



/// '//'/ / / 
////// / //////*'<< 

2 

/z 7 // 

// /^7/ /////// 

/^y-GRAZING ANCLE / y / 

' // 

\r t 

MiSSM 

mmw 



V-REFRACTED RAY 



Figure 10. Geometry of reflection and refraction. 


For horizontal polarization, the electric vector for 
the incident wave then is 

F t = Eoe 


where ^ is the grazing angle, / the frequency of the 
radiation and c the velocity of light in free space. 
The electric vector of the corresponding reflected 
field is given by the similar expression 

E r = Eo pe~ J<t> ~ j2ir * 1* "d 1 / c) cos ^ + z sin Wl Q 0 ) 

where p and <fr are real constants. The ratio of the 
reflected to the incident field at the reflection point 
{z = x = 0) is seen to be 

R = (17) 

By definition this is the reflection coefficient for 
horizontally polarized waves. Thus the reflection 
coefficient is a complex quantity, the amplitude of 
which is the reflection coefficient of the wave 
amplitude and the phase is the lag in phase of the 
reflected wave with respect to the incident wave at 
the point of reflection. 

The reflection coefficient for vertically polarized 
radiation is defined in the same way. It is found, 
however, that the expression of p and <j> in terms of 
the grazing angle ^ and the ground constants 
are quite different for the two types of polarization. 
For an arbitrary position of the plane of polariza¬ 
tion, the wave must be separated into its vertically 
and horizontally polarized components, and the 
proper reflection coefficient applied to each compo¬ 
nent separately. 

The quantities p and <f> are determined by the 
boundary conditions for the electric vector at the 
reflecting surface, namely, that the tangential 
components of the electric vector on the two sides 
of the boundary surface shall be equal. This brings 
in the ground constants, that is, the conductivity 
and dielectric constant of the reflecting body. How 
these boundary conditions are applied may be 
illustrated by the simple example of horizontally 
polarized rays reflected from a surface of infinite 
conductivity. In the surface itself, the sum of the 
incident and reflected field strength must always 
be such that the currents set up in the body just 
suffice to produce the reflected field. Within a 
reflecting body of infinite conductivity, an infinitely 
weak field is sufficient, and hence the boundary 
condition is such that the reflected field, at the 
reflection point, shall be equal in magnitude and 
opposite in phase to the incident field, so that the 
resultant field is zero. Hence for infinite conduc¬ 
tivity and horizontal polarization 

, P = 1, 0 = 180°. (18) 


,j2nf [*-(l/c) (a; cos \f/-z sin <^)] 


(15) 


R = — 1 












54 


FACTORS INFLUENCING TRANSMISSION 


424 Fresnel’s Formulas 

For finite conductivity, the reflection coefficient 
may assume a variety of values. The general formu¬ 
las, as derived from electromagnetic theory, are 
given in equations (19) and (20). For horizontal 
polarization 

R = sin ^ — V e c — cos 2 )/ ' (jg) 

sin tf/ + V ( c — cos 2 )/' ’ 


ization the phase change is 180° from \f/ — 0 up 
to an angle \J / 0 determined by tan ^o=l/Ve f . 
Here, tfie coefficient is zero. For larger angles the 
phase change is zero. As e r increases indefinitely, 
ypo approaches zero and for infinite e r the phase shift 
is zero everywhere, except for \J/ = 0 where it is 
indeterminate. The angle \J /o is called the Brewster 
angle. For \p = 0 the amplitude is unity, and for 
i = 90° 


and for vertical polarization 

€ c sin if — V c c 


R = 


COS 2 if 


( 20 ) 


e c sin \p + V — cos 2 if ’ 

where e c is the complex relative dielectric constant of 
the reflecting ground which is given by 

= € f - j . (21) 

A material that acts like a good conductor for 


Ve, ~ 1 

v ^+1 


( 22 ) 


for both cases of polarization. When € t - is no longer 
zero, the amplitude p will show a deep minimum 
for a certain value of \ff instead of the zero found for 
€i = 0. The angle corresponding to the minimum is 
called the pseudo-Brewster angle. These various 
points are illustrated by examples in Figure 16. 



Figure 11 . Dielectric constant of sea water at 17 C. 

low-frequency waves may act as an approximately 4 2 5 The Complex Dielectric Constant 
pure dielectric for microwaves. The case e,- = 0 is of Water 

therefore of considerable practical importance. When As much of the available radar and communica¬ 
nt = 0, and R consequently is real, the phase lag is tion equipment is either shipborne or erected along 
180° for horizontal polarization. For vertical polar- the coast, reflection from sea water is one of the 





































GROUND REFLECTION 


55 


principal problems to be discussed here. For micro- 
waves the salt content in sea water makes little 
difference, so that it may be assumed that the dielec¬ 
tric constant and conductivity are the same over all 
oceans at the same temperature. With increasing 
temperature, the real part of the dielectric constant 
diminishes roughly by one unit per 5° C. Figure 11 
gives the dielectric constant e r of ordinary sea 
water at 17 C as a function of frequency. 

The dielectric constant also diminishes with in¬ 
creasing salinity, but in the UHF-SHF region, normal 
variations of salinity have much smaller influence 
than changes in temperature and frequency. The 
imaginary part of the dielectric constant is, for 
frequencies less than say 1,000 me, related to the 
conductivity a as follows: 


An average for inland lakes is <r = 10 “ 3 mho per 
meter. For wavelengths shorter than about 10 cm, 
the dielectric constant is influenced by the fact that 
water is built up of polar molecules. The maximum 
effect is found for wavelengths of the order of 1 cm. 
For this region, there is no appreciable difference 
between salt and fresh water. 

The probable run of € t - for sea water is shown in 
Figure 11. The part of the e; curve extending from 
6,000 me to 30,000 me corresponds essentially to 
results obtained at Clarendon Laboratory, Oxford. 
Other investigators give results which are markedly 
different. This curve is thus affected by consider¬ 
able uncertainties and should be used with caution. 

In Figures 12 to 15 are shown amplitude and 
phase of the reflection coefficient for a smooth sea for 


€i = + 60 <r\ (23) 

(<r in mhos per meter and X in meters). At 25 C the 
average conductivity of sea water is usually given as 



Figure 12. Amplitude of the reflection coefficient p 
versus angle of reflection 'i' for sea water. (From 
Radiation Laboratory Report C-ll.) 


4.3 mhos per meter. The temperature dependence is 
given by 

<7 = (725 [1 + 0.020 - 25)], 

where t is temperature in degrees centigrade. 

The conductivity of fresh water is much smaller. 



Figure 13. Phase of the reflection coefficient 0 versus 
angle of reflection for sea water. Reflected wave E r 
lags incident wave E t - by 0. (From Radiation Labora¬ 
tory Report C-ll.) 

various frequencies. Figure 12 gives the amplitude 
p of the reflection coefficient for both kinds of 
polarization, for reflection angles up to 90 degrees, 
and for 100 and 3,000 me. Figure 13 gives the 
phase 0 under the same conditions. The next two 
figures give p and 0 on a greatly enlarged scale of 
0, in the interval 0 = 0 to 0 = 5.5°, which embraces 
all cases of practical interest. 








































































































































56 


FACTORS INFLUENCING TRANSMISSION 



Figure 14. Amplitude p of the reflection coefficient 
versus reflection angle ^ from ^ = 0 to ^ = 5.5° for 
sea water. 



Figure 15. Phase <p of the reflection coefficient versus 
reflection angle ¥ from V = 0 to ^ = 5.5° for sea water. 

E r lags E» by </). (From Radiation Laboratory Report 
C-ll.) 

42 6 Overland Transmission 

Conditions over land are very different from those 
found over the sea. Land as a reflecting surface has 
larger irregularities and their effect is more pro¬ 


nounced. Therefore in selecting a radar site, it is 
preferable to choose a location which is surrounded 
by relatively smooth ground. The electrical proper¬ 
ties of the earth vary considerably for different 
localities, so that it is necessary to study the ground 
conditions for each particular case. Experimental 
data concerning reflection of very short waves from 
ice- or snow-covered ground seem to be lacking. 
Precise information might be of operational interest, 
particularly in Arctic regions. Laboratory exper¬ 
iments indicate that ground covered by ice or snow 
will influence the propagation of short waves some¬ 
what in the same way as very dry ground. 

4 2 7 Conductivity of Soil 

Extensive investigations have been made on the 
conductivity of different types of soil, particularly 
on low and medium frequencies. For 10 me, the 
observed values range from 6 • 10” 3 mhos per meter 
for chalk to 0.13 mhos per meter for blue clay. The 
conductive increases with increasing moisture 



Figure 16. Amplitude of the reflection coefficient for 
moist and dry soil. 


content, so that marked seasonal changes may be 
anticipated for a given locality. It also varies with 
frequency. Under field conditions it will not be 
possible to measure the conductivity in individual 
cases and one will have to assume a value of about 








































































































































GROUND REFLECTION 


57 


10" 2 mhos per meter for poorly conducting ground 
like chalk or very dry soil and take a value of about 
10 _1 for good conductors like blue clay or water- 
bogged marshy land. Fortunately, the amplitude 
of the reflection coefficient is not very sensitive to 
minor changes in conductivity when the frequency is 
sufficiently high, say 200 me or higher. Then the 
real part of the dielectric constant is the most 
important factor. 


Dielectric Constant of Soil 


It is not possible to give a standard table of 
dielectric constants of various types of soil, because 
the variation with the moisture content is consider¬ 
able. For very dry ground e r is likely to be about 4, 
but this value may rise to 25 when the ground is 
thoroughly soaked with water. The dielectric con¬ 
stant of ground will normally decrease with increas¬ 
ing frequency. 

Above 200 me, the dielectric constant will dom¬ 
inate the conductivity term, and for field conditions 
the ground may be assumed to be a pure dielectric. 
This is illustrated in Figure 16 for e r = 7; e t = 3 





-T - ! 

i -1 

POLARIZA 

II 

ei. 




-r- 








''K | 

\ y 

- 

P MOIST 
>r*25,€ 
\ 

SOIL / 
•= 19.2 / 

1 ^ 







\ 

\ 

y r =7. 

V 

’e-3 DRY 

SOIL 






\ 

\ 

\ 

\ 

\ 

\ 








l 

1 








VERTIC/ 

L POLAR 

IZATION 








l 

l 

\ 

\ 








\ 

\ 

\ 

\ 

\ 

\ 








\ 

\ 

\ 

s' 

A 










' \ 


— 







f 







K? 20 s 30 5 40 5 SO 3 60° 70° 80° 90° 


Figure 17. Phase of the reflection coefficient for 
moist and dry soil. 


and €i = 0; and for e r = 25, = 19 and e,- = 0. 

Except for values close to the Brewster angle, the 
zero conductivity curves give a usable approxima¬ 
tion. 


In Figure 17, the phase, <£, of the reflection coeffi¬ 
cient corresponding to the above values of e r and e t 
is also given. 


4 2 9 The Divergence Factor 

The preceding considerations apply only to 
reflection from plane surfaces. For reflection from a 
sphere like the earth, the divergence of a bundle of 
rays is increased when it suffers reflection, and the 
plane earth reflection coefficient, R, must be multi¬ 
plied by a divergence factor denoted by D, which 
accounts for the earth curvature. This factor ranges 
from unity at close range where the earth can be 
considered plane to zero at points just above the 
tangent line. [Note: When the divergence factor 
approaches zero at grazing angles less than the last 
minimum, other components of the wave must be 



Figure 18. Geometry for divergence factor. 


considered.] To a sufficient approximation D is 
given by the expression 




2 hfhf I'* 
dka tan 3 \{/ J 


(24) 


where (see Figure 18), 

hi, hf = heights of transmitter and receiver above 
tangent plane at reflection point. 
d = distance between transmitter and re¬ 
ceiver measured along the surface of 
the earth. 

4/ = grazing angle above tangent plane. 
ka = equivalent earth radius. 


4.2.10 Irregularity of Ground 

The formulas for reflection from a plane or a 
spherical earth can only be applied with confidence 
granting a certain smoothness of the reflecting 
surface, depending on the wavelength. A rule of 





























58 


FACTORS INFLUENCING TRANSMISSION 


thumb for the applicability of the reflection formulas 
is that the vertical height of the irregularities should 
not exceed X/16^, where X is the wavelength and^ 
the grazing angle in radians. Suppose, for instance, 
that the wavelength is 1 meter and the grazing angle, 
1 degree. The limit of tolerance is then 56/16 = 3.5 
meters. Hence, on this wavelength one may expect 
specular reflection over sea in most cases. For 
X = 10 cm, on the other hand, the limit is only 35 cm, 
and for X = 3 cm it is 11 cm. For larger grazing 
angles, the limit of tolerance will be correspondingly 
smaller. 

4 3 DIFFRACTION (GENERAL SURVEY) 

431 Definition 

The term diffraction will be understood to apply 
to those modifications of the field produced by 
material bodies outside the transmitter that cannot 
be described by the ray methods of geometrical 
optics. 

With this limitation of the term diffraction, there 
are three main topics to be considered: 

1. Diffraction by the earth’s curvature. 

2. Diffraction by irregular features of the terrain, 
such as hills, houses, etc. 

3. Diffraction by objects, primarily metallic 
(targets) in two-way transmission (radar echoes). 
Also scattering by raindrops. 

4.3.2 Diffraction by Earth’s Curvature 

The diffraction field in this case is the field appear¬ 
ing below the line of sight determined by use of the 
equivalent value of the earth’s radius ka. The case 
of an idealized earth with smooth surface and given 
electrical properties can be treated mathematically, 
and the field obtained is often designated as the 
standard field (see Chapter 5). If one moves away 
from the transmitter horizontally, at a fixed height 
above the earth, the field strength decreases expo¬ 
nentially with distance once the line of sight is 
passed. Similarly the field strength decreases expo¬ 
nentially with height above the ground on going 
vertically downwards from the line of sight. In 
many instances, the variation in the field strength, 
in the diffraction region is independent of the electric 
properties of the ground. The main exception occurs 


in a comparatively shallow layer near the ground. 
Only for the important case of propagation over sea 
water and for frequencies below 100 megacycles 
does this layer become high enough to cover an 
appreciable part of the whole diffraction region. 

4 3 3 Diffraction by Terrain 

The problem of diffraction by terrain features 
requires special treatment. Frequently a field of 
appreciable magnitude is found behind hills, houses, 
etc. Diffraction is also important when there is a 
sudden change in ground properties, as for instance 
in a transition from land to sea. In this case the 
shore line acts as a diffracting edge. Only a limited 
number of cases lend themselves to evaluation by 
simple formulas. The cases are those which can be 
treated by the Fresnel-Kirchhoff method of optics 
which leads to a somewhat intricate but straight¬ 
forward mathematical formula for the diffracted 
field strength. In spite of its apparent limitations, 
the Fresnel-Kirchhoff formula is often applicable 
to short-wave propagation problems. It is treated 
in Chapter 8. 

4 34 Diffraction by Targets 

This problem can be dealt with theoretically by 
methods similar to those used in computing diffrac- 



Figure 19. Reflecting pattern of an airplane. 


tion by terrain features, the main difference being 
that the angle of scattering is nearly 180 degrees 
instead of approximately 0 degrees. 

In the case of target diffraction, theory is less 






DIFFRACTION (GENERAL SURVEY) 


59 


useful than in many other problems of wave propa¬ 
gation. This is due to the fact that radar targets 
such as airplanes or ships have an extremely complex 
structure; the scattered intensity will therefore 
often change by many decibels as a result of only a 
small tilt of the target. Figure 19 shows a typical 
reradiating or reflecting pattern for an airplane. 
Numerous measurements of the average radar cross 
section of planes and ships have been made. 

A phenomenon of great importance in the micro- 
wave region is the scattering of radiation by water 
drops in the atmosphere. Small droplets such as are 


found in fogs and most types of clouds do not give 
reflection visible on the scopes. Only drops large 
enough to produce actual precipitation give appre¬ 
ciable radar echoes. However, this does not neces¬ 
sarily mean that rain is falling at the locality indi¬ 
cated by the scope; frequently vertical updrafts of 
air will maintain drops afloat that in still air would 
fall to the ground; moreover, drops falling from a 
comparatively high cloud can evaporate before 
reaching the ground. Especially in tropical regions, 
the last-named phenomenon is more common than 
is ordinarily thought. 



Chapter 5 

CALCULATION OF RADIO GAIN 


51 INTRODUCTION 

5,1,1 Objectives 

T his chapter is devoted to the definition and 
calculation of the various factors which enter 
into a computation of the field strength of radio 
waves propagated in the standard atmosphere above 
the earth. 

In Chapter 2, particularly in Sections 2.1 and 2.2, 
are given the basic definitions of path-gain factor 
and radio gain for transmission between doublets 
and other antenna types in free space. The present 
chapter shows how these quantities must be modified 
to account for the influences introduced by the 
curvature and electrical properties of the earth. 

The methods of computation are presented in 
considerable detail to enable the interested reader to 
apply them to his particular problem, and sample 
calculations are given which should assist in reducing 
to a minimum the time required for obtaining the 
answers in a given case. 

5,1,2 Definitions Relative to Radio Gain 

The radio gain is defined as the ratio of received 
power P 2 , delivered to a load matched to the receiver 
antenna, to transmitted power Pi, with both an¬ 
tennas adjusted for maximum power transfer. For 
doublet antennas in free space this ratio is given by 
(3X/87T d) 2 and is denoted by A 0 2 , that is, 



and the free-space gain factor is given by 


reflection and diffraction effects of the ground are 
taken into account, the expression for the radio gain 
becomes a very complicated affair. 

For the general case of one-way transmission, the 
radio gain is given by 

§ = G 1 G 2 A i , (3) 

Pi 

where Gi and (7 2 are the gains of the transmitting and 
receiving antennas, respectively, and A , the gain 
factor, is equal to 

A = A„A p (4) 

with A p equal to the path-gain factor. [See equation 
(27) in Chapter 2.] 

For radar or two-way transmission, the radar gain 
is decreased because the energy traverses the path 
both ways and is influenced by the radiating proper¬ 
ties of the target as given by the radar cross section <r. 
Combining equations (46) in Chapter 2 and (2) in 
this chapter, the radar gain equals 

g -TO 

Comparing equations (3) and (5), it is seen that the 
gain factor A may be used also for two-way trans¬ 
mission, provided the additional term 167r<r/9X 2 is 
included in the formula. 

Later on it will be shown how to split up A and A P 
into a product of various factors, represented by 
graphs which make it possible to carry out computa¬ 
tions in specific cases. 

The gain factor A may also be expressed in terms 
of the field strength E and free-space field strength of 
a doublet transmitter E 0 . From equation (28) in 
Chapter 2, 


A 0 


3X 

Sird 


( 2 ) 


in which d denotes the distance from the transmitter 
to the receiver measured in the same units as the 
wavelength X. 

When the radiation is emitted and received by 
directive antennas and the propagation takes place 
through a refracting and absorbing atmosphere, and 


E = E 0 ylG 1 A p . (6) 

Combining this equation with equation (4) 

±.S-L, m 

a, «.Vc, 

where Eq^Gi is the free-space field of the trans¬ 
mitting antenna with gain G\. 


60 


INTRODUCTION 


61 


The free-space field, in terms of the transmitted 
power Pi, is given by 

'JChEo = ( 8 ) 

d 

for a point in the direction of maximum radiation. 
In terms of the power P 2 delivered to the load circuit 
of a receiving antenna, with matched load and 
oriented for maximum pickup, the field at any point in 
space is equal, from equation (17) in Chapter 2, to 

A ' (jf 2 

It is sometimes convenient to express E in terms of 
the (radiation) field at one meter from the trans¬ 
mitter, whence E 0 = E x /d and, from equation (6), 

* = f Vg1a # . do) 

5.1.3 Factors Affecting Attenuation 

and Gain 

The above definitions are quite general. In the 
absence of the earth, there remains only the free- 
space attenuation which results from the spreading 
out of the radiated energy as it moves away from 
the transmitter. At a distance which is several times 
larger than the wavelength, the field strength varies 
inversely as the distance from the antenna. 

The presence of the earth affects the field through 
two sets of quantities. One set is geometric and 
includes the heights of the antennas and their dis¬ 
tance apart, the curvature of the earth, and shape 
of terrain features. The other set is electromagnetic 
and depends on the dielectric constant and con¬ 
ductivity of the earth and of its atmosphere, the 
polarization and the wavelength of the radiation. 

5.1.4 Simplifying Assumptions 

The present chapter is mainly concerned with the 
computation of the field-strength distribution of a 
transmitter for certain idealized standard condi¬ 
tions, so chosen as to give a fair average picture 
of propagation conditions for very high-frequency 
radiation. The reasons for this limitation are stated 
in Chapter 1. In substance, the limitations are 
imposed by the great complexity of the general 
problem, which makes it necessary to proceed in 
successive steps. The first step is to consider propa¬ 


gation under standard conditions, which will be 
defined farther on. Successive steps take into 
account diffraction by terrain, that is, by trees, 
hills, mountain ranges, or shore lines, or by non¬ 
standard propagation effects in the atmosphere. 

The fundamental importance of a knowledge of 
propagation under standard conditions is first of all 
due to the fact that in a large number of cases condi¬ 
tions do not differ significantly from standard. On the 
other hand, when they do deviate significantly, the 
standard solution sets up a criterion for the discov¬ 
ery of deviations and the evaluation of the influence 
of the nonstandard conditions upon propagation. 

The basic assumptions which define what we have 
been calling standard propagation conditions will 
now be given. 

1. Standard atmosphere. It is assumed that the 
index of refraction of the atmosphere has a uniform 
negative gradient with increasing elevation. As has 
been pointed out in Chapter 4, the influence of such 
an atmosphere upon propagation is equivalent to 
that of a homogeneous atmosphere over an earth of 
radius ka, where k is a constant that usually is taken 
equal to 4/3. 

2. Smooth earth. The earth is assumed to be 
perfectly smooth. It can be considered sufficiently 
smooth if Rayleigh’s criterion is satisfied, that is 
when the height of surface irregularities times the 
grazing angle (in radians) is less than X/16 (see 
Section 4.2.10). 

3. Ground constants. The dielectric constant and 
conductivity of the earth are assumed uniform. For 
wavelengths less than one meter this assumption is 
particularly valid since in this case propagation is 
largely independent of the ground constants. In 
the YHF (1 to 10 m) range, the same is true with the 
important exception of vertically polarized radiation 
over sea water. For the VHF range, the assumption 
of uniform earth constants is unsatisfactory for 
paths partly over land and partly over sea water, or 
over sea water with large land masses near-by (see 
Chapters 8 and 10). 

4. Doublet antenna and antenna gain. For the 
formulas of this chapter, the radiating system is 
assumed to be a doublet antenna (i.e., a straight 
wire, short compared to the wavelength). Actual 
antennas have radiation patterns different from 
that of a doublet, usually having greater directivity. 
The antenna gain of a half-wave dipole is 1.09 times 
(or 0.4 db greater than) that of a doublet, the field 
maximum being the same in the two cases. This 





62 


CALCULATION OF RADIO GAIN 


gain is insignificant in practice. For other types of 
antenna systems and for microwave frequencies, the 
gain may be many times larger. 

The propagation problem, thus limited, has been 
solved mathematically; but the explicit mathematical 
formulas are far too complicated to be of much use to 
the practical computer. Much additional work has 
been done, however, to bring the solution into a 
form suitable for practical use. This involves 
reducing the computations to the use of graphs, 
nomograms, and tables, and it is this final stage of 
the problem which is the subject of subsequent 
parts of this chapter as well as of Chapter 6. 


equation (11) assumes the form 

-is?- 

or 

d T = V2 kahi . (13) 

Similarly, the horizon distance of the receiver is 

d R = V2 kah 2 . (14) 

The sum of the two horizon distances is given by 
d Lj where 

d L = d T d R . (15) 


Curved-Earth Geometrical 
Relationships 


Let hi and h 2 denote the heights of transmitter 
and receiver above the earth’s surface, respectively, 
and let d denote the distance from the base of the 
transmitter to the base of the receiver, measured 
along the earth’s surface. For a number of cases 
concerned with high-frequency radiation over the 
earth’s surface, it is sufficient to identify the straight- 
line distance from transmitter to receiver with the 
distance d between the bases measured along the 
curved earth. But when path differences are of 
importance, as they are in interference problems of 
reflection and in diffraction, it is necessary to com¬ 
pute distances to a higher order of accuracy. 

Throughout this chapter, the earth will be assumed 
to have the equivalent radius ka, and the atmosphere 
to be homogeneous, and radiation to travel along 
straight lines. 

The straight line from the transmitting antenna 
and tangent to the earth’s surface (the so-called 
line of sight) touches the earth along a circle which 
constitutes the radio horizon of the transmitter. 
The distance measured along the earth from the 
transmitter to the radio horizon will be denoted by 
and the horizon distance of the receiver by d R . 
These geometrical relations are illustrated in 
Figure 1. 

From this figure, it follows that 

ka + hi =-—- . (11) 

cos( d T /ka) 


Inasmuch as 
ka 


ka 


cos( d T /ka) 


- (. d T /ka) 2 


2 ka 


5,1,6 Optical and Diffraction Regions 

The points visible from the transmitting antenna 
(on an earth of equivalent radius ka), i.e., the points 
above the line of sight, constitute the optical region 
(Figure 1). The rest of space lies beyond the trans¬ 
mitter horizon and below the line of sight and is 
called the shadow or diffraction region. 


OPTICAL REGION 
OPTICAL HORIZON 



Figure 1 . Geometry for radio wave propagation over 
curved earth. 


It is frequently necessary to know whether a 
receiving antenna lies in the optical region or the 
diffraction region of a given transmitter. This 
evidently is equivalent to knowing whether the 
distance d of the receiver from the transmitter is 
smaller or larger than the combined horizon distance 
d L . By equations (13), (14), and (15), it follows 
that in the optical region 

d < V2to (VftT+ V^), (16) 

and in the shadow region 

d > V2 ka (ylhi + V/i 2 ). 


(17) 










INTRODUCTION 


63 


A graphical representation of the equation 

d L = V2 ka (V/q + Vfc) (18) 

is given in Figure 2. For k = 4/3, a = 6.37 X 10 6 m, 
this takes the form 

d L = 4120 (V/ix + ylh^) meters, (19) 
where h h h 2 are given in meters and d L in meters. 


h, METERS 

Or— 


25 

100 


200 

300 

400 

600 


800 

1000 


2000 

3000 

4000 

5000 

6000 

7000 

8000 

9000 

10000 


d L km 

Or— 

100 — 

200 — 

300 — 

400 — 

500 — 

600 — 

700 — 

800 — 

900 — 


h 2 METERS 
Or— 

25 — 

100 — 


200 — 
300 — 
400 — 


600 

800 

1000 


2000 

3000 

4000 

5000 

6000 

7000 

8000 

9000 

10000 


15000 


20 000 


1000 

1100 

1200 


15000 


20000 


d u = 4.l2 (/*b VM’d T + d R 


Figure 2. Sum of transmitter and receiver horizon 
distances for standard refraction. To change scale: 
Divide hi and h 2 by 100, and divide d^ by 10. 


517 Nature of the Radiation Field 
in the Standard Atmosphere 

The mathematical solution for the radiation field 
takes various forms for particular cases. The treat¬ 
ment for low antennas, for instance, differs from 
that for high antennas, and similarly the equations 


must be handled differently for the two types of 
polarization. These and related problems are dis¬ 
cussed in general in the following. 

1. General form of field variation. The mathemat¬ 
ical expression for the radio gain of the radiation 
field of a doublet under standard conditions is given 
as the sum of an infinite number of complex terms or 
modes. (See Section 5.7.6.) Disregarding the phase 
factor, a representative term (mode) of this series 
has the form 

Fid) • Mh) • f 2 (h 2 ) . (20) 

These modes are attenuated unequally. Well within 
the diffraction region, the first mode contributes 
practically all of the field so that the effects of dis¬ 
tance and height are separable. In this region, the 
problem of numerical computation is simplified, 
since it is possible to use separate graphs for the 
dependence on height and distance. As the receiver 
is moved toward the transmitter, the number of 
modes required for a good approximation increases. 
For low antennas, the addition of the modes is 
practicable and the graphical aids are useful for 
short distances. These conditions are illustrated in 
Figure 3 for horizontal polarization or ultra-short 
waves. 

In the optical region, the methods of geometrical 
optics give a result equivalent to that of the rigor¬ 
ous solution at points which are not close to the line 
of sight. The field is then the sum of a direct and a 
reflected wave, resulting in an interference pattern. 

The preceding discussion is illustrated by Figure 4 
which shows the variation of field strength with 
distance for fixed antenna heights, for propagation 
over dry soil with a wavelength of 0.7 meter on 
vertical polarization. The numbers refer to the 
number of modes required for a better than 99 per 
cent approximation. The interference pattern is 
illustrated by the oscillatory nature of the curve. It 
will be observed that beyond the first maximum, 
the points found by geometric optics give a value 
of the field which is slightly too low. (See dots in 
Figure 4.) In fact, as the line of sight is approached, 
the optical formula approaches zero whereas the exact 
solution does not. The geometric-optical method 
breaks down in the optical region as the line of 
sight is approached. It may be noted that Figure 4 
has been drawn for k = 1 rather than for the cus¬ 
tomary value of k = 4/3 corresponding to standard 
atmosphere conditions and is for a hypothetical 
isotropic radiator. 







64 


CALCULATION OF RADIO GAIN 


transmitter low 


OPTICAL-INTERFERENCE REGION 



TRANSMITTER ELEVATED 


OPTICAL-INTERFERENCE REGION 



Figure 3. Field regions as related to transmitter heights for horizontal polarization or ultra-short waves. Low antenna 
region = \d > 2 hji 2 ‘, h x , hz < 30X 2,3 . 



Figure 4. Variation of field strength with distance for propagation on vertical polarization with a wavelength of 70 cm 
-over dry soil. The point “due to minimum” results from minimum in reflection coefficient at the pseudo-Brewster angle. 













































































INTRODUCTION 


65 


If the earth were flat and perfectly reflecting, the 
envelope of the maxima of the curve in Figure 4 
would coincide with the line 2 E 0 , twice the free- 
space field, corresponding to the in-phase addition 
of the direct and reflected waves. An envelope of the 
minimum points would be E = 0, corresponding 
to the destructive interference of the direct and 
reflected waves. The curvature of the earth, resulting 
in increased divergence of the waves (see Section 
5.2.5), and the lack of perfect reflection (see Section 
5.2.4) cause the maximal and minimal envelopes to 
differ from 2 E 0 and 0, respectively. In the neighbor¬ 
hood of the first maximum in Figure 4 (i.e., when the 
direct ray makes small angles with the earth), the 
reflection coefficient tends to be unity in magnitude 
for both polarizations except for the increase in diver¬ 
gence which results in the deviation of the maxi¬ 
mal and minimal lines from 2 E and 0, respectively. 
At a smaller distance, for vertical polarization, as 
shown in Figure 4, the deviation is caused principally 
by the smaller magnitude of the reflection coeffi¬ 
cient. The virtual meeting of the maximal and 
minimal lines corresponds to the minimum value 
of the reflection coefficient at the pseudo-Brewster 
angle. (For horizontal polarization at small dis¬ 
tances, the envelope of maxima would virtually 
coincide with 2 E 0 and the minima would be closer to 
zero. As the distance is increased, the difference 
between the envelopes for vertical and horizontal 
polarization gradually decreases.) 

2. Both antennas low; h < h c . In a discussion of 
the height function, it is convenient to distinguish 
between high and low antennas. The critical height 
separating the two cases for horizontal polarization 
or ultra-short waves is given by 

h c = 30\ 2/3 meters (21) 

where X is expressed in meters. For X = 0.1 meter, 

h c = 6.46 meters, and for X = 10 meters, h c = 139.5 
meters. If both antennas are at elevations less than 
h c , the height-gain functions f(h), to a first approx¬ 
imation, are the same for all the modes, so that the 
complete solution 

Mh) . fm . fm) + f 2 (h) . f 2 (h 2 ). F 2 (d) + + + 
can be written in the form 

(Fi + ft + (22) 

or 

/(/M)-/(^)-F(d), (23) 

where f(hi) replaces fi(hi), f(h 2 ) replaces fi(h 2 ), etc., 

while F(d) stands for the sum Fi + F 2 + • • • . 


The distance function F(d) can be calculated for 
particular cases. This has been done for high fre¬ 
quencies and is represented graphically in Section 
5.7, the results being valid for low antennas for all 
distances in the optical as well as in the diffraction 
region such that 2hji 2 < < \d (see Figure 3). The 
condition 2 hih 2 << \d assures that the antennas are 
below the interference pattern. 

At the ground, f(h) = 1, so that if both antennas 
are close to the ground, the distance dependence is 
given by F(d) only. 

3. One or both antennas elevated; h> h c = 30X 2/3 . 
For elevated antennas, h > h c and the height-gain 
functions of f(h) vary v r ith the modes. Conse¬ 
quently, it is not possible to separate the height and 
distance effects as in the previous paragraph. 

In the optical-interference region, it is more ad¬ 
vantageous to use the method of combining the 
direct and reflected waves. This is equivalent to 
the rigorous solution which is illustrated by the dots 
in Figure 4. 

Simple graphical aids can be given for points well 
within the diffraction region where the first mode 
predominates. The range of usefulness of the first 
mode can be extended by plotting the field strength 
given by the first mode as a function of height (or 
distance) and plotting a similar curve by using the 
ray method as far as the lowest (or first) maximum 
(see Figure 7). Then by joining these partial curves 
into a smooth overall curve, a fairly good value of 
the field can be obtained for intermediate points. 

There is a further possibility occurring with the 
transmitting antenna elevated, the receiver low and 
lying below the interference pattern, and the dis¬ 
tance short. In this event, none of the previous 
methods apply. However, the reciprocity principle 
(see Chapter 2) can be applied to find the radio 
gain at the receiver by interchanging the role of 
receiver and transmitter. Suppose the original 
transmitter height is 100 meters, the original re¬ 
ceiver height 15 meters, and the wavelength 1 meter. 
Now let the transmitter height be 15 meters. If the 
receiver height is low ( h 2 < 30 meters), values of 
the gain can be found (Section 5.7); if the receiver 
height is in the interference region, the gain can be 
found by the ray method. Now suppose a curve be 
drawn for these results, giving the attenuation 
versus receiver height. From this graph, the value 
of the gain factor A at h 2 = 100 meters can be read. 
This value of A by the principle of reciprocity is the 
gain factor for the original heights. 



66 


CALCULATION OF RADIO GAIN 


4. Ultra-short waves in the diffraction region. 
Dielectric earth. For X < 10 meters (/ > 30 me) and 
for either polarization, land acts as a dielectric earth 
or absorbing earth in contradistinction to a conducting 
earth. Propagation over a dielectric earth is practi- 



Figure 5. Field strength ratio versus distance for 
vertical polarization over dry soil for h j = 100 meters 
and h-i = 0. 



Figure 6. Field strength ratio versus distance for 
vertical polarization and heights h x = h 2 = 100 meters. 


cally independent of earth constants. For a given 
type of polarization, the chief variables affecting 
gain are then the heights of the antennas, their 
distance apart, and the wavelength. Within the 
diffraction region, the effect of increasing wavelength 
is to increase the field strength. This is illustrated 
by the curves in Figures 5 and 6. The dielectric earth 
is characterized by a value of 8 > > 1. 8 is given 
by equation (193). 

While sea water has a relatively high conductivity, 
radio wave propagation over it is the same as that 
over a dielectric earth in the case of horizontal polar¬ 


ization for X < 10 meters, and in the case of vertical 
polarization for X < 1 meter. Consequently, verti¬ 
cally polarized radiation of wavelength range 1 to 10 
meters over sea water is given special treatment in 
Section 5.7.4. 

In the same range, 1 to 10 meters, for vertically 
polarized radiation and for distances less than those 
given in Table 3, propagation conditions over land 
also deviate slightly from those corresponding to a 
dielectric earth. 

5. Optical region. In the optical-interference 
region, the lobes for the shorter waves are more 
numerous, narrower, and lower, as can be seen from 
the oscillatory part of the field strength versus 
distance curves of Figure 6. 

The dependence of reflection coefficients upon 
polarization, wavelength, and ground constants is 
discussed in Section 4.2. 

6. Horizontal versus vertical polarization. In the 
optical region, for rays at small grazing angles, there 
is not much difference between the two types of 
polarization. For larger grazing angles, the differ¬ 
ence is more marked (see Section 4.2 on reflection 
coefficients, and see Section 5.2.4). 



Figure 7. Gain versus height at distances beyond the 
radio horizon. 

It has been pointed out in the previous paragraph 
that within the diffraction region for X < 10 meters 










































































PROPAGATION FACTORS IN THE INTERFERENCE REGION 


67 


and propagation over land, there is practically no 
difference in intensity between a horizontally and a 
vertically polarized radiation field. For X < 1 meter 
there is, similarly, no difference for propagation over 
sea water. When there is a difference, as for low 
antennas, horizontal polarization gives a smaller 
gain; but as the antennas are raised, the two cases 
approach equality (see Section 5.7.4). 

52 PROPAGATION FACTORS 

IN THE INTERFERENCE REGION 

521 Propagation Factors 

The factors affecting gain in the region where the 
methods of geometrical optics may be applied are 
discussed in Sections 5.2 to 5.5 inclusive. 

5 2 2 Spreading Effect 

From the formula of equation (1) in Chapter 2, 
for the field intensity components of the radiation 
field, it follows that for distances from the transmit¬ 
ter large in comparison to the wavelength, the domi¬ 
nant term falls off inversely as the distance from the 
transmitter, or 

E=^, (24) 

d 

where Ei is the field strength at unit distance. This 
means that the power per unit area in the radiation 
field varies inversely as the square of the distance. 
This spreading effect is the consequence of the fact 
that the energy of the wave is distributed over larger 
and larger areas as the wave progresses away from 
the transmitter. 

523 Interference 

When a wave travels over a conducting surface, 
constructive and destructive interference occurs 
between the direct wave from the transmitter and 
the wave reflected by the surface. This is illustrated 
in Figure 8, which is drawn for a plane earth. If 
there is no energy lost in reflection, the direct and 
reflected waves are of equal intensity, and their 
resultant varies from zero to twice the free-space 
value, depending upon the phase difference between 
the two components. The reflected wave lags the 


direct wave by an angle 5 + <£, where S is the phase 
retardation caused by the greater path length 


R 



earth. 

traversed by the reflected wave and </> is the phase lag 
occurring at reflection. 

Figure 9 shows the vector diagram for the case 
where the phase shift at reflection is 0 = 180 degrees. 


E r 



Figure 9. Vector diagram showing the addition of 
the direct and reflected waves for </> = 180° and p = 1. 

This condition holds for horizontally polarized 
radiation of frequency above 100 megacycles, re¬ 
flected from sea water at grazing angles of less than 
10 degrees. The resultant electric field is equal to 

E = -J E<? + E* - 2E 0 E r cos 6 

= ^(E 0 - E,Y + 4 E 0 E r sin 2 1 . (25) 

If the reflection is complete, as from a conductor of 
infinite conductivity, 

E r = E 0 , E = 2E 0 sin — . (26) 

2 

5 2 4 Imperfect Reflection 

In general, the strength of the reflected wave E r 
is less than that of the incident wave E { , partly be¬ 
cause of diffuse reflection and partly because some 
energy is refracted into the surface and absorbed. 















68 


CALCULATION OF RADIO GAIN 


Furthermore, the phase lag usually differs from 
180 degrees, depending upon the frequency and 
grazing angle. This is especially true for vertical 
polarization where the reflection coefficient is a 
critical function of both the grazing angle and fre¬ 
quency. The ratio 

R = ^ = pe'* (27) 

Ei 

is a complex number and defines the reflection co¬ 
efficient R, which has an absolute value p and a 
phase angle </>. 

In equation (27), a lagging angle is considered 
positive. Writing </> = tt + 0', equation (27) may 
be expressed as 

R = — = pc’*” + *' ) = - pe(28) 

Ei 

In equation (27), the lag angle 0 is measured with 
respect to zero-degree phase shift at reflection. For 
horizontal polarization, 0 varies from 180 degrees to 
183 degrees, and from 180 degrees to 3 degrees for 
vertical polarization at 3,000 me over sea water. 
In equation (28), lag angle 0' is measured with 
respect to a 180-degree phase shift (that is, from 
Ei reversed), and varies from 0 degree to 3 degrees 
for horizontal polarization and from 0 degree to 
—177 degrees for vertical polarization. 

The resultant field intensity is 

E = E 0 + pE^-M +♦> = E 0 + p£’oe-' (S+ *' + ’ r) . 

= E a (l - pe - ja ), 

where 

12 = 5 + 0 r = 5 + 0 — 7r. 

The absolute value of the received field | E | is 
given by 

|S| = £„^(l-pe + ^) (1 -pe-* a ) 

or- 

| E | = E 0 ^ 1 + p 2 — 2p cos 12 

= ^o A J(l-p) 2 + 4psin 2 |. (30) 

Equation (30) shows that the received field intensity 
has a maximum of 2 E 0 when 


In equations (31) and (32), n includes all integral 
values and zero. Equation (30) may be written as 

E = (1 -p) 2 + 4psin 2 |. (33) 

where E is the field at distance d from the trans¬ 
mitter, and Ei is the field strength at unit distance. 
From equation (33), 

d = •%!(!- p ) 2 + 4 #> sin2 | • ( 34 ) 

In free space where there is no reflecting earth, 
p = 0, and 

d„ = |i, (35) 

hi 

where d 0 is the equivalent free-space distance from 
the transmitter at which the field strength E would 
be found. Hence equation (34) may be written in 
the form 

d = d °yJ(l -p) 2 + 4psin 2 |- (36) 


5,25 Divergence 

The divergence factor D is introduced to account 
for the decreased gain produced by the spreading 



Figure 10. Increased divergence resulting from re¬ 
flection by a sphere. 


P = 1, 

12 = (2 n + 1)tt. 

The value of E is a zero when 


(31) 


of a wave reflected from a spherical surface, 
ferring to Figure 10, 


p = 1, 


E& 

lda 2 _ 

1 da 2 da 0 ^ 

£1 

II 

1 O 1 

IJ5LL 

12 = 2mr. 

\y*>) 

E 2 

^d^ 3 ~ 

' dao da s r 3 

^ da 3 r 3 


Re- 


07) 

















GENERAL SOLUTION 


69 


In calculating the field intensity reflected from a 
spherical earth, the inverse distance attenuation 
factor 1 /r d used for the direct wave must be multi¬ 
plied by the divergence factor D, which is always 
less than unity. 

As a result of this divergence the reflection 
coefficient for a spherical surface is less than that 
for a plane surface as given by 

P D = P ' (38) 

where p' is the spherical earth reflection coefficient. 
Equations (30) and (36) may then be written as 

|£| = £ on )(1 _ p ')«+ Vsin’l (39) 

and 

d = d 0 -yj (1 — p') 2 -f- 4p' sin 2 • (40) 

5 -2.6 Antenna Gain and Directivity 

The effects of antenna gain and directivity are 
expressed by means of the gain factor G, defined in 
Section 2.2.2, and the antenna pattern factors F i 
and F 2 , which are the fractions of the maximum radi¬ 
ation amplitude in the direction of the direct and 
reflected rays respectively. The maximum ampli¬ 
tude for a transmitting antenna with gain G\ is 
Eo = VGi Eo, where E 0 is the free-space field 
strength radiated by a doublet. 

When the antenna pattern factors are taken into 
account, the resultant field intensity, following 
equation (30), is 

E = <G X Eo V E? + - 2F l !<\ P D cos 0. (41) 



Figure 11. Antenna pattern factors F i and F 2 . 

The factors Fi and F 2 are functions of the angles 
7 i — a and v + a which the direct and reflected 
rays make with the axis of the beam. Figure 11 is a 
typical antenna pattern. 


53 GENERAL SOLUTION 

5,3,1 Generalized Reflection Coefficient 

Equation (41) may be simplified by introducing a 
generalized coefficient which includes the effects of 
reflection, divergence, and directivity. The ampli¬ 
tude of this coefficient will be denoted by K and is 
given by 

K = ~ pi). (42) 

U 1 

Substituting F 2 = FiK/pD in equation (41) gives 

E = J(hFiE 0 V 1 + K 2 - 2K cos 0 (43) 

or 

E = ylGlFiEoJ (1 - K) 2 + 4 K sin 2 - . (44) 

2 



Figure 12. ^(1 — K ) 2 + 4 K s k l2 -^- as a function of 
K and 12. 


If the transmitting antenna is pointed so that the 


direct ray lies in 

the direction 

of 

maximum 

gain, 

Fi = 1, 

and 







E 

= ylGlEo 

J( i- 

- A) 2 + 

4 K 

sin 2 

fi 

2 

(45) 

and 








d 

= 1 

s/u- 

- Ky + 

4 K 

sin 2 

fi 

9 * 

(46) 






































































70 


CALCULATION OF RADIO GAIN 


Figure 12 shows 

yj (1 - K)* + iK sin 2 - 

2 

as a function of K for various values of sin £2/2 and 
ma} r be used to calculate E. 

The value of Gi to be used in equation (46) may 
be found from the antenna specifications for a given 
set. The free-space field E 0 at distance d from a 
transmitting doublet wdth power output Pi is equal to 


E 0 


3V5 VPj 
d 


(47) 



Figure 13. Free-space range d as a function of field 
strength E 0 and transmitter power P x . 


Figure 13 shows E 0 in decibels above 1 microvolt 
per meter as a function of d for various values of 
transmitted power. The free-space field at distance 
of d meters expressed in decibels above 1 microvolt 
per meter for Pi watts radiated is 

Decibels = 20 logio \ . (48) 

L lCT 6 d J 


54 PLANE EARTH 

541 Use of Plane Earth Formula 

It wall be shown in Section 5.5.5 that under cer¬ 
tain conditions calculations based upon the assump¬ 
tion of a plane earth yield satisfactory results. Cal¬ 
culations for propagation over a plane earth are 
given in this section. 


Path Difference 


It follows from equation (29) that when p = 1 
and </> = 180 degrees, the received field depends 


only upon the phase lag caused by path difference. 
Referring to Figure 8, 


r d 


= yjd^+(h l - h,y- = dyj 1 + (> 


(49) 


r = yjd* + (h + h*Y = d yjl + , 

(5C 

. ,r2 hh, hMhi'+hf), 1 

i -r-r.-d|_—---+ -..J. 

_ 2Kh,V _ + W 

2d 2 + “J’ 


If 


2d 2 

hS + A 2 2 


2d 2 


<< 1, 


2hih- 2 


d * 


(51) 


(52) 


The phase lag caused by the path difference A is 
equal to 

^ _ 2Tr/2hih 2 \ 4rhih 2 


X \ d / Xd ’ 
where X is the wavelength of the radiation. 


(53) 


5 4 3 Field Strength Equations 

When p = 1 and <j> = 180°, equation (26) may be 
used. Substituting equation (53) for 8 into equation 
(26) gives 

E = 2 E, sin . (54) 

If 8/2 < 10°, sin (5/2) 8/2 and 

(55) 

Xd 

When hi or h 2 equals 0, equation (55) indicates that 
the received field intensity is equal to zero, which is 
contrary to fact. For this case the diffraction field 
must therefore be calculated and included as ex¬ 
plained in Section 5.1.7. 

In the general case (p < 1), equation (44) may 
be applied with K = (P 2 /Pi) pD. A refinement may 
be added to the calculation by taking into account 


































































SPHERICAL EARTH 


71 


the fact that the image source (Figure 8) of the 
reflected wave is at a distance r + A from the re¬ 
ceiver. The reflected wave is attenuated more than 
the direct wave, according to the free-space attenua¬ 
tion ratio (r + A)/r. If this is taken into account 
the ordinary reflection coefficient is replaced by 



The correction is not necessary when 2h x h 2 << d 2 
[see equation (52)]. 

55 SPHERICAL EARTH 


In order to express the slant range r d in terms of the 
curved distance d to a higher order of accuracy, the 
cosine law is applied to the triangle, transmitter- 
receiver-earth center. This gives the equation 
rf = (ka + h x ) 2 + (ka -f h 2 ) 2 

— 2 (ka + hi) (ka + h 2 ) cos ( — ) . 

\kaJ 

Selecting the relatively important terms of the order 
d 2 hih 2 and d*(h x + h 2 ), as well as powers of d higher 
than the fourth, the above equation reduces to 

ri = <P + (h - /ii) 2 + f-(h l + h 2 - —-). (57) 
ka \ Vika/ 


551 Measurement of Distance 

The difference between the slant range r d and the 
distance measured along the surface of the earth and 
designated by d in Figure 14 is usually negligible. 
For a transmitter height of 1,500 meters, the error 
in assuming r d — d is 0.04 per cent at a distance of 
161 km and height of 6,900 meters, and 1 per cent 
at the same distance but at a height of 22,500 meters. 
As the transmitter height is increased, the error is 
increased. 


2 Equivalent Heights 

Solving equations (13) and (14) for h x and h 2 gives 

2 ka 


These results may also be expressed by saying that 
the distance from the surface of the earth to a plane 
which is tangent at a distance di from the trans¬ 
mitter is d x 2 /2ka. 



Figure 14. Geometry for radio wave propagation over a spherical earth. (Vertical dimensions greatly exaggerated.) 








72 


CALCULATION OF RADIO GAIN 


Hence for a transmitter of height hi above the 
ground the height above the tangent plane at the 
reflection point, the so-called equivalent height is, to a 
first approximation, 

di 2 


so that 


hi' = h- 


2 ka * 


and for the receiver the equivalent height is 

d*_ 

2 ka 


h 2 ' = h 2 - - 


(58) 


(59) 


The equivalent heights are shown in Figure 14, 
which illustrates the geometry of the spherical earth 
having an effective radius of ka. 

55,3 Angles 

Referring to Figure 14 and remembering that the 
angles are greatly exaggerated in the figure, it is 
seen that 



* - 5 (1 - &), 
z 

(66) 

and 

'T? 

1 

t-H 

+ 

II 

e* 

(67) 

where 

h _ di - d 2 

(68) 


di -j- d 2 

and 

^ hi — h 2 

(69) 


hi + h 2 


Assume hi > h 2 and di > d 2 , so that h and c will 
always be positive. This is always possible because 
of the principle of reciprocity. From equation (60), 

W _ W or hi _ di _h 2 _ d 2 
di d 2 di 2 ka d 2 2 ka 

Substituting for di and d 2 from equations (64) and 


tan \f/ 

~ hi' _ 
d ., 

11 

(60) 

(66), 

hi+ h 2 j 

f l + c\ 

d( 1 + b) 

tan y 

~ h*_ 

d 

d 

2 ka’ 

(61) 


d \ 

hi + h 2 j 

a + h) 

a - c\ 

4 ka 

d( 1 - b) 

tan ypd 

~h 2 - 

hi d 

(62) 


d \ 

<1 -b) 

4 ka 

d 

2 ka 1 

Simplifying, 




V 

^ * + 

di 

ka 

(63) 


hi + h 2 

2 (c - b) 

bd 





d 

1 - b 2 

2 ka 


Angle \[/ must be evaluated in order to determine the 
reflection coefficient. Angles \f/d and v determine the 
antenna pattern factors F x and F 2 , which are shown 
in Figure 11. The angle y is significant in coverage 
calculations and angular approximations. 


Solving for c, 


where 


c = b + hm( 1 — b 2 ), 


m ■ 


d 2 


5.5.4 


Determination of Reflection 
Point (dj 


Aka(hi + h 2 ) 


(71) 

(72) 


Inasmuch as several equations of Sections 5.5.2 
and 5.5.3 depend upon d h it is necessary to be able 
to determine this distance when the transmitter 
and receiver heights are given and the distance 
between them is known. Let 


di = - (1 + b), 
Z 


(64) 


and 


To determine di, equation (71) must be solved 
for b. This is a cubic equation, which is easily solved 
when rn is small in comparison to unity. However, 
for m values comparable to unity, or larger, it is 
easier to plot a series of curves showing c as a func¬ 
tion of m for assigned values of b ranging from 
0 to 1. These are straight lines with a slope of 
5(1 — b 2 ) and are given in Figure 15. 

The procedure for calculating di when hi, h 2 , and 
d are given is as follows: 


hi = (1 + c), (65) 


1. Compute c = 


h- 


hi + h 2 















SPHERICAL EARTH 


73 


2. Compute m = -. 

4 ka(hi -f- h%) 

3. Read b from Figure 15. 

4. Calculate di = — (1 + b). 

2 

It should be noted that hi may be the height of 
either the transmitter or receiver. The only re- 


From equation (70) 

hi_ _ _sd_ _ h 2 _ (1 — s)d 
sd 2 ka (1 — s)d 2ka 

Simplifying, 

, 3 „ T ka„ . , » 1 ”1 , kahi _ 

S 2 S S L^ l + A2) -2j + ^ = °' (74) 



Figure 15. Graph for obtaining b for given values of c and m. (Marconi, Ltd.) 


strictions are that hi > h 2 and di represents the 
distance from hi. 

Another method will now be given for calculating 
di. This method will be particularly useful when 
di/d < < 1 and will be applied in Section 5.5.8 on 
generalized coordinates. Let 

di = sd, s < 1. (73) 


If s < < 1, the terms s 3 and s 2 may be neglected 
to a first approximation, and 


s 


hi 


(hi + h 2 ) — 


d 2 
2 ka 


(75) 


If d < < d L (Section 5.1.5), h 2 is well above the line 
of sight and s reduces to 


hi _ di 

hi -f- h 2 d 


Hence 


c? 2 — (1 — s)d. 


( 76 ) 












































































74 


CALCULATION OF RADIO GAIN 


Equation (76) is the plane earth formula. 

Curves showing s as a function of h 2 /hi and 
d/drp are given in Figures 19 and 20. These may be 
used for the direct calculation of di = sd within the 
limits of interpolation. 


Path Difference 


Referring to Figure 14, the path difference A 
for a spherical earth is equal to 


A = r — r d = Vd 2 + ( h 2 + hi)* 

-Vd 2 + (W - W)\ (77) 

It is usually sufficient to expand the square roots 
and neglect powers and products of h\ and h 2 
beyond the second. This gives 


A = r — r d 



(78) 


which is the same as the plane earth formula when 
hi and h 2 are written instead of hi and h 2 . Equation 
(78) is accurate to within 1 per cent for values of y 
(the angle at the base of the transmitter) less than 
about 8 degrees. The error is less than 10 per cent 
for values of y less than about 24 degrees. When 
equation (78) is not sufficiently accurate, the follow¬ 
ing may be used: 

2W 

A — r — r d = 7 ~ - > (79) 

Vd 2 + (wy + (wy 

provided 

1 wy+iwy <<x 

2 d 2 


All the above equations for the path difference 
depend upon the distance to the reflection point di. 
However, the calculation of di may be eliminated 
by first computing the path difference from the 
plane earth formula and then subtracting the correc¬ 
tion term A(A,). Thus 


where 


and where 


A = A p — A(Ap), 
2hih 2 


Ap — 


d 


550 A / A X 

- A (Ap) 

/ l ! 3 ' 2 

is given in Figure 16, plotted against 
d d 1 

4 ~ V2 ka ' 


(80) 


If h 2 < h h interchange h 2 and hi on the curves and 
ordinate of Figure 16. 

The maximum value of A (A p ) is 

AM— .0.33X10-^=^' < 8 ‘> 

If the plane earth correction factor is negligible for 
the wavelength under consideration, the plane earth 
formula may be used throughout the whole range 
within the optical region, not only for the given value 
of h 2 /hi but for all lower values of h 2 /hi with the 
same hi. 

When h 2 >> h h so that the reflection point is 
much closer to the transmitter than to the receiver, 
a good approximation to A is obtained by replacing 
hi by hi and h 2 by h 2 — d 2 /2ka in equation (78). 



A ^ 



-) 

2 ka/ 


f 


which, to the same approximation, means that 


A = 2 hi tan y. 


(82) 


In general, equation (82) is an improvement over 
the plane earth approximation except close to the 
transmitter and at low heights where h 2 is not much 
greater than hi. 




































































SPHERICAL EARTH 


75 


The dependence of path difference upon distance 
and height may be seen by considering the path 
difference parameter 


R = 


ka A 
hi dj> 


(83) 


Since h 2 = Wcb/di, it follows from equations (78) 
and (59) that 


= 2W = 2 (h'ydt = 2ck h , 

d ddi d 


(x _ JL\ 

\ 2 kahj m 


di 


Hence, using d T 2 = 2kahu 


a _2VM\ [1 ~ (di. 

dj \d/ di/ 


- { di/d T yY 
Jd T 


and 


R = d, x [1 -{di/d T yr 
d d\f dj< 

= A _ di/d T \ [i -{di/d T y? 
\ d/d T ) d\/dj< 


(84) 


(85) 


The form of this expression suggests the introduction 
of two new dimensionless parameters 

p = — and v = — . (86) 

d T d T 


In terms of these parameters, equation (85) for R 
assumes the lorm 



(i - P >y 

1 

(87) 

\ v! 

and in terms of s and v 

P 


R = (1 - s) 

(1 - $V) 2 

sv 

(88) 


Using equation (60), D becomes 
1 


If d 2 - 


D = 


D = 


V 1 + 2di 2 d 2 /kahi'd 


V1 + 2 hi/ka tan 2 \f/ 


(7 < 3°) 


and if \p is small, so that tan \J/ — >\J/, 
1 

D = -7====. 
V 1 + 2 W/kai 2 


(90) 

(91) 


(92) 


Parameters p and q 


Useful expressions for the divergence factor, path 
difference, and receiver height may be obtained by 
use of the dimensionless parameters, 


and 


or 


di 

V ~ V2 kahi 


A 

dj< 


(93) 


q = 


d 2 

~d 


(94) 


di — (1 — q)d = sd. (95) 


The divergence factor may be expressed directly 
in terms of p and q by modifying equation (90) as 
follows: 



Vl + 2d^/kaWd 

1 


4 


1 + 


4(d 2 /d) (di 2 /2kahi) 
1 — (di 2 /2kahi) 


5 5 6 Divergence Factor 

The reflection of a beam of radiation from the 
spherical earth increases the divergence of the beam 
and reduces the intensity of the reflected wave by 
spreading, as explained in Section 5.2.5. This is 
taken into account by introducing a divergence 
factor D, less than unity, which appears in the formu¬ 
las as a multiplier of the plane earth reflection coeffi¬ 
cient. Expressions for D are 


V1 -j- 2hi%'/ kad tan 3 \f/ 


where W has been replaced by its equivalent ex¬ 
pression, given in equation (58). The above form 
of D shows that it can be expressed in terms of 
p and q only: , 

D = - ( - 1 -- • (96) 

Vl+ 4p 2 q /(1 — p 2 ) 

Figure 17 shows contours of constant D as a function 
of p and q. 

The path difference A may be written in terms of 
p and q by substituting into equation (78): 

A _ 2W _ 2{hi) 2 d 2 _ 2d 2 h 2 (l—di 2 /2kahi) 2 
zx — — ri\ ■ • 

d ddi d di 






























76 


CALCULATION OF RADIO GAIN 



Figure 17. Divergence factor D as a function of p and q or p and s. (Radiation Laboratory.) (p = 1 is the line of sight 
where D = 0.) Note: S = 1 — q = djd. 


Hence, on using equations (93), (94), and (95), 


A = 


2/q yi -py 

d r p 


(97) 


Figure 18 shows (1 — p 2 ) 2 /p as a function of p. 

The receiver height h 2 may also be expressed in 
terms of hi, p, and q. This will be found useful in 
drawing coverage diagrams in which both h 2 and d 


are unknowns. From equation (60) 


or 


W = V 

di d 2 

2 ka di \ 2 kaj 







































































SPHERICAL EARTH 


77 


Hence it follows that 



Since 

(I 2 _ d 2 _ q 
di (1 — q)d 1 - q ’ 

*• = -»(*)( + !*)• » 



The distance from the transmitter to the reflection 
point di and the ratio p may be eliminated by using 
the dimensionless coordinates 



V = ^ . (100) 

dj< 

The advantage of this substitution lies in the fact 
that the coefficients of s = d\/d in the cubic equation 
(74) may be expressed as functions of u and v only. 


Thus, 


3 3 
s 3 - 

2 


ka 
~d 2 


(/?i + h 2 ) 


os _ 3 _ 1 r (i + w 

2 2 L d‘/(2kah l ) 




a 


+ 


hi n 
+ ka~~0, 


2d 2 / (2kahi) 


0. 


In terms of u and v , 

5 3 _ 3 s2 _ £ / i ±^_ 1 \ 1 = 0 

2 2 \ ?.’ 2 ) W ’ 


Figures 19 and 20 show contours of constant s 
plotted in u, v coordinates. The curves are parab¬ 
olas. 



Figure 19. s as a function of u and v. (See Figure 14 
for definition of lengths.) 



Figure 20. s as a function of u and v. (See Figure 
14 for definition of lengths.) 







































































































































































78 


CALCULATION OF RADIO GAIN 


The divergence factor D is given by 


D 


where 

and 



_1_ 

Vl-f-4p 2 g/(l — p 2 ) ’ 

p = sv 



( 102 ) 


(103) 


Since s is a function of u and v only, it follows that D 
may be plotted in u , v coordinates. This may be 
accomplished by solving equation (96) with respect 
to q, which gives 


(1- P 2) (1-2)2) 
4 p 2 D 2 


(104) 


Equation (103) gives s = 1 — q; v = p/s, and u 
may be read from Figure 19 or 20. Contours of 
constant D are shown in Figures 21, 22, and 23. 

The grazing angle ^ is important in calculations 
for vertical polarization since it determines the 
magnitude and phase of the reflection coefficient 


for a particular frequency and reflecting surface. 
The grazing angle \f/ may be expressed in terms of 
s, u, and v, as follows. From equation (60), 


tan ^ S — = — (1-^—) . 

di di \ 2kahi J 


(105) 


Hence 

tarn/' 

or 


h 


(1 - s 2 


d T (sd/dT ) 

tan \p 1/1 


a 2 ) = j- (- - w ) 
d,T \sv / 


V hi V2 ka 

and for k = 4/3, 
tan ^ 

-j=r = 2.4 X 10- 

V hi 




(106) 


(107) 


From Figure 24, \f/ may be obtained for given values 
of hi and sv = p. 

The generalized coordinates described in this 
section will be found highly useful both in field 
strength and coverage calculations. 



Figure 21. Contours of constant divergence factor D and path difference variable R. (Radiation Laboratory.) 






























CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 


79 



Figure 22. Contours of constant divergence factor D and path difference variable R. (Radiation Laboratory.) 


56 ILLUSTRATIVE CALCULATIONS FOR 
THE OPTICAL-INTERFERENCE REGION 

561 Introduction 

The general expression for the gain factor A in the 
interference region is obtained by combining equa¬ 
tions (44) and (7). Then 

A = AoFt y](l- KY + sin 2 f • (108) 

The value of the radio gain is then given by equa¬ 
tion (3) and the value of radar gain is given by equa¬ 
tion (5). The value of the radical which defines the 
interference pattern has a range of values between 
0 and 2. The extreme values can occur only when 
K — 1 (p = 1, D = 1, F 2 /Fi = 1); the value 0 
(nulls) is then given by sin 2 (ft/2) = 0 and the value 
2 (maxima) is given by sin 2 (ft/2) = 1. 

In general, the value of A lies between the two 
extremes 

A = A 0 F l (1 ± K), 


the positive sign giving a maximum and the negative 
giving a minimum. At any other point, the value 
lies between these two extremes. For range calcula¬ 
tions (which involve maxima), the variation in A 
is from 1 to 2 times the free-space value, according 
to the value of K, so that in practice a quick rule of 
thumb for range may be devised. Assume (1 + K) 
equal to 1.9 for favorable conditions (sea water, 
horizontal polarization, or, in the case of vertical 
polarization, small grazing angles) down to (1 + K) 
= 1 or K = 0 for propagation over rough terrain. 
The problem of finding the range is thus reduced 
to a problem for free space. In range calculations, 
P 2 /P 1 is given by the ratio of minimum detectable 
power to power output. A is then determined by 
equations (3) or (5), and the range is given by find¬ 
ing d from the relation (writing A 0 = S\/8t d) 

a -(£) <i+k> - (m) 

More detailed calculations are presented in this 








































































80 


CALCULATION OF RADIO GAIN 



Figure 23. 


Contours of constant divergence factor D and path difference variable R. (Radiation Laboratory.) 


section. However, the assumption is made generally 
that the reflection coefficient is equal to —1 (i.e., 
p = 1 and (j) = 180 degrees) and that the direct and 
reflected rays do not differ appreciably on account 
of the shape of the antenna beam pattern (F 2 = Fx). 
For large distances over sea water, these assumptions 
are approximately realized. For most of the calcula¬ 
tions it offers no inherent difficulty to consider the 
effect of directivity or of a reflection coefficient 
different from — 1 but may require considerable 
additional calculation (see Sections 5.6.2 and 5.6.5). 

For convenience, the formulas required in the 
calculations are recapitulated here. Putting p = F x 
= F 2 = 1, equation (108) takes the form 


A = A 0 yj (1 - D) 2 + 4 D sin 2 1- (110) 

or, in decibels, 

20 log A — 20 log Ao+101og|(l- D) 2 + 4 D sin 2 -^-1. 


The reflection point variable 


d\ d\ 
dr V 2kahi 

= - - 1 - (when k = 4/3) 

(4120 V/ii) 


(HI) 


will be used extensively. It has been found that the 
interference pattern is very sensitive to slight changes 
in p, so that an accuracy to the fourth significant 
figure is generally required. 

The path difference variable R is related to p — d\/ d? 
and v = d/dx by the equation 

b = (---) d - p 2 ) 2 . (H2) 

\p v / 


Resolved with respect to \/v , this equation assumes 
the form 


l = l/l_ Ry \_l-g 

v p\ (1-p 2 ) 2 / p 


( 113 ) 






















































CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 


81 


Another convenient expression for R is obtained by 
replacing, in equation (83), the path difference A 


.03 

.04 

.05 


.1 


.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


.98 


^(degrees) h, (METERS) 

r— lOVz 


75 

60 

45 

30 

20 

10 

5 



.2 



.02 


.01 

.0 05 

.002 


.001 



Figure 24. xp as a function of h x and p = sv=di/d T . [See 
equation (107).] (See Figure 14 for definition of lengths.) 


by nX/2, where n (see below) may assume any posi¬ 
tive value. Substituting V 2kahi for d T , equation (83) 
assumes the form 


where 


R = nr, 


(114) 


1 I ka X 

2^~2hF 2 


or 


r = 1030 —Y (for A; = 4/3). 

h\ 


A graphical representation of r is given in Figure 15 
in Chapter 6. 

Then for a reflection coefficient of p = 1, <f> = 180 
degrees (i.e.,0' = 0), equation (29) gives 

Q = 5=—A = nr. (116) 

X 


If r is fixed, a complete pattern of contour lines 
(along which A is constant) is determined. Take as 
independent variables p and n (rather than u and v). 
A given choice of p and n determines R by equation 
(114), £2 by equation (116), v by equation (113), 
s by equation (118), u by equation (121), D by 
equation (117), and finally 20 log A by equation 
(110). By varying r, new patterns are obtained. 
Accordingly, r may be called a pattern or chart 
parameter (see Section 6.8.3). 

The lobes on the charts depend on n, in accordance 
with equation (116). Accordingly, for p = 1, 
</> = 180 degrees, n is the lobe variable. For the 
first (lowest) lobe, n — 0 gives the first null, n — 1 
gives the first maximum and n — 2 the second null. 
For the second lobe, n varies from 2 to 4, with a 
maximum at n = 3, and so on. It should he remem¬ 
bered that if n < 1, corresponding to the lower side 
of the lowest lobe, the value of the field (or of A) given 
by the optical formida is too low. A more accurate 
value can be obtained by joining the curves found 
in the optical and diffraction regions into a smooth 
overall curve. 

Combining equations (102), (103), and (113) gives 
the divergence factor D, 



+ 77 


4 Rp< 


(i - p 2 )- 


r-t 


The variable s = d\/d is 


+ 


4:Rp 2 d 2 *l - 1/2 

1 - p 2 ’ ~d J 

(117) 


* = vK 

and, repeating equation (101), 



(118) 

(119) 


In terms of p, equation (119) becomes 

2 p 3 — 3 p 2 v + p(v 2 — u — 1) + v = 0. (120) 

Equation (120), resolved for v and u, gives 


i r i / 

(l 

Y 

» = i 3p--+ \M 

( ” P 

) +4 u\ 

2L p y 

\p 

) J 


u = 2p 2 - 3pv + - - 1 + V 2 . 
V 


(115) 


( 121 ) 










82 


CALCULATION OF RADIO GAIN 


Some useful approximations: 


For v 

large, q — » 1 and R = — (1 
V 

— p 2 ) 2 ap- 

proaches the value 



R~--2p, 

V 

(122) 

which, solved for p, becomes 



-R+ Vfl 2 + 8 

P ~ - . 

4 

(123) 

If R > 2, 

1 

v ~ —. 

R 

(124) 

If R < 2, 



3 -R 
v~ 4 • 

(125) 

For R > 3, 

D ~ 1. 

(126) 


The calculations will be divided into four types. 

Type I. The direct calculation of the radio gain 
(or field) when the heights and distance apart of the 
antennas and the wavelength are given. 

Type II. The calculation of the radio gain as a 
function of the receiver height h 2 when the trans¬ 
mitter antenna height h h distance d, and wavelength 
X are given. 

Type III. The calculation of the radio gain as a 
function of the distance d when the transmitter 
antenna height hi, receiver height h 2 , and wavelength 
.are given. 

Type IV. The calculation of the possible positions 
of the receiver in space (h 2 ,d) when the radio gain 
will have the given value for given values of the 
gain factor A, the transmitter antenna height hi, and 
the w'avelength X. Special cases, such as the re¬ 
ceiver antenna height, h 2 for given d, or d for given h 2 , 
can be solved by use of the curves in Type II and 
Type III, in Sections 5.6.3 and 5.6.4. 

This type of problem is of importance in estimat¬ 
ing the range of a set when the minimum detectable 
power of the receiver and power output of the trans¬ 
mitter are knowm. 

5.6.2 p ro kl em G f Type I. Radio Gain 
for Fixed Heights and Distance 

The radio gain at a given receiver is to be found, 
their heights, as well as the wavelength being given. 


The polarization is assumed horizontal, the 
effective earth’s radius 4a/3. 

The following data are assumed. 

Transmitter height: hi = 50 meters 
Receiver height: h 2 = 1,500 meters 

Distance apart: d = 100 kilometers 

Wavelength: X = 1 meter (/ = 300 me) 

Gains (over doublet): Gi = C 2 = 100 (20 db) 

One-Way Transmission 

1. d L = 188 kilometers (Figure 2). d < d L , so 
that the receiver is in the optical region. 

2. The u,v coordinates of the receiver are 

u = ^ = 30, 

h 50 

d T = y/2kahi = 29,100 meters 



d T 29.1 


3. Referring to Figure 19, s ( = p/v) is estimated to 
be 0.05. Since the result is very sensitive to slight 
changes in s, it is desirable to improve the value of s. 
In Newton’s method, the next approximation, using 
equations (101) or (119), is 



s' = 0.04794. 

The next approximation gives the same result. 


4. Using the above value of s' and the relation 
p = sv (; v = 3.43), p is equal to 

p = 0.1645. 

With the value of p = 0.1645, equation (117) and 
Figure 22 give the value 

D = 0.95. 

5. The path difference variable R is obtained from 
equation (112) or Figures 21, 22, 23, as 

R = 5.477. 

6. The number r, from equation (115), is 2.91, so 
that the lobe number n is 

n = - = 1.88. 
r 

Hence the receiver is on the upper part of the first 
lobe close to a null. 










CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 


83 


7. ft = nr = 5.9 [see equation (116)], 



sin 2 - = 0.0353. 

2 

8 . To use equation (110), the value of the free- 
space gain factor A 0 is needed. Figure 3 in Chapter 2 
gives 

20 log A 0 = - 118. 

Substituting in equation (110), 

20 log A = 20 log A 0 + 10 log (0.0025 + 0.1342) 

= -118 —8.64 == -127 
By equation (3), using gains of 20 db, 

10 log — = -127 + 40 = -87. 

P 1 

Accordingly the radio gain is — 87 db and the received 
power is given by 

p 2 = pao" 8 - 7 . 

Suppose the receiver has a minimum detectable 
power of 10“ 10 watt, then the required minimum 
power output under the given conditions would be 

Pi = P 2 X io 8 - 7 

= 10“ 10 X 10 8 ' 7 = 10 “ L3 watts. 


Radar 


Suppose that, instead of a receiver, there is a target 
at the same position with a radar cross section of 
a = 50 square meters. The value of P 2 /P 1 at the 
radar receiver can be found from equation (5) using 
the value of A found above and the given values 
of cr and X. If the radar uses the same antenna for 
transmitting and receiving, Gi = C 2 , which in this 
case is 100 or 20 db, and the radar gain 


10 log — = 20 log Gi + 10 log + 10 log a 
Pi 9 


+ 40 log A - 20 log X, 
= 40 + 7.5 + 17 - 254 - 0, 

= - 189.5 db 


This gives Pi = 10 8 ' 9 watts, which obviously is unat¬ 
tainable. 


ft in equation (108) is no longer given by equation 
(116) but is the sum of two phase shifts, one caused 
by path difference, (P/r) 7r = m, while the other is 
<£'=</> — 7 r, the difference between the phase of the 
reflection coefficient and that for perfect reflection. 
Hence 

ft = — 7T + </> — 7T. (129) 

r 

The lobe variable (for imperfect reflection) is 
now N, defined by 

ft = Nt (130) 

rather than n = R/r. The relation between N and n 
is derivable from equation (129), giving 

N = n — . (131) 

7 r 

The propagation is assumed to take place over 
sea water. The angle between the reflected wave 
and the earth is given by equation (107) or Figure 24, 
and is 

¥ = 0.582°. 

From Figures 14 and 15 in Chapter 4, 

</> = 168°, p = 0.76. 

The lobe variable N, in terms of the old lobe variable 
(for p = 1, (j) = 180°), by equation (131), is 

N = 1.88 - — = 1.81, 

180 

- = 163°, 

2 

sin 2 — = 0.085. 

2 

The fact that N < n signifies that the lobe for 
vertical polarization (other things being equal) has a 
greater angle of elevation than the lobe for hori¬ 
zontal polarization. 

K = P D = 0.76 X 0.95 = 0.722 

The value of 10 log £(1 — K) 2 + 4 K sin 2 -^ J 

= 10 log 0.322 = — 5. 


Effect of Vertical Polarization 

The general value for K in equation (108), when 
the reflection coefficient p differs from — 1 and 
F 2 /F 1 = 1 is (see Section 5.3.1) 

K = P D. (128) 


Therefore, for vertical polarization, 

20 log A = -118 - 5 = -123, 

which may be compared with the value 20 log A 
= —127, obtained for horizontal polarization with 

p = 1. 




84 


CALCULATION OF RADIO GAIN 


5 6.3 Type II. Radio Gain Versus 

Receiver Height for Given Distance 

Radio gain versus receiver antenna height are 
to be found, while transmitter antenna height, 
wavelength, and distance are given. 

Suppose that a radar set has an antenna height 
of 30 meters, and an antenna gain of 13.5 db. Polari¬ 
zation is horizontal and the wavelength is 1.5 meters. 
Assume also a receiver with a gain of G 2 = 1 (or 0 db) 
at a distance of 100 km. 

The following calculations are made: 

1. The variation of the radio gain P 2 /Pi, with 
receiver antenna height h 2 is to be found. 

2. Instead of a receiver assume a target with 
cross section of 50 square meters. The value of the 
radar gain P 2 /Pi at the radar receiver is to be found 
as a function of target height h 2 . 

The diffraction part of the calculation is given in 
Section 5.7.3; the optical part in this section. The 
results are represented in Figure 25, the two partial 
curves for one-way transmission having been com¬ 
bined into a smooth overall curve which makes pos¬ 
sible the estimation of 10 log P 2 /P 1 in the transition 
region near the line of sight. The radar gain varies as 
40 log A [equation (5) ] rather than as 20 log A and 
contains a constant shift 10 log [(?i 2 (167ro-/9X 2 )] 
rather than 10 log GiG 2 . 

Radio Gain: One-Way Transmission 

The calculation is most readily performed by using 
p = di/d T as the independent variable and then 
finding the corresponding values of h 2 , the receiver 
height, and A. 

1. From Figure 15 in Chapter 6 or equation (115), 
r = 9.403. 

For n — 1, corresponding to the first maximum, 
p is approximately 1/r, since R = nr > 2. Accord¬ 
ingly, we begin with p = 0.1. 


2. dr = 22.5 km (from Figure 2), and 

d 100 

v = — =-= 4.45. 

dj> 22.5 

3. From equation (112), 

R = 9.58. 

4. From equation (121), 

u — — 62.028 

h 

and hence 

h 2 = 1,861 meters. 

5. At d = 100 km, the free-space attenuation 
(Figure 3 in Chapter 2) is 

20 log A 0 = - 115. 

6. Compute the factor 

>j(l - D) 2 + 4J9 sin 2 —. 

2 

From equation (117) 

D = 0.980, 

n =- = 1.019, 
r 

ft = nir = (1.019) • (3.1416) = 3.19 radians, 
12 

- = 1.60 radians, 

2 

sin 2 - = 0.9992. 

2 

From Figure 12, 

20 log ^ (1 — D) 2 + 4 D sin 2 = 6 db. 

Hence 

20 log A = — 115db + 6db = — 109 db, and 

10 log — = - 109 db + 10 log CnGi = - 95.5 db. 
Pi 

7. The foregoing values, together with results 
obtained with other values of p, are listed in Table 1. 


Table 1* 


V 

R 

s 

h 2 

n 

Q 

2 

D 

t 

Radio Gain 
P 2 

10 log ^ 

Radar Gain 

P 2 

10 log ^ 

0.15 

6.16 

0.0337 

1,396 

0.655 

1.03 

0.96 

5 

-96.5 

-172 

0.1 

9.58 

0.0225 

1,861 

1.019 

1.60 

0.98 

6 

-95.5 

-170 

0.05 

19.68 

0.0112 

3,216 

2.09 

3.29 

0.995 

-11 

-112.5 

-204 

0.04 

24.70 

0.00898 

3,885 

2.63 

4.13 

0.997 

4 

-97.5 

-174 

0.03 

33.05 

0.0067 

5,005 

3.51 

5.52 

0.998 

3 

-98.5 

-176 

0.02 

49.74 

0.0045 

7,235 

5.29 

8.31 

0.999 

5 

-96.5 

-172 

0.01 

99.76 

0.0022 

13,917 

10.61 

16.67 

1.000 

4 

-97.5 

-174 


* See also Table 5. 


t 20 \og^ (1 -D) 2 -f- 4D sin 2 (0/2). 










CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 


85 


The value of P 2 /P 1 [equation (3)] is represented 
graphically in Figure 25. 



Figure 25. Radio gain in decibels versus height h 2 . 
Horizontal polarization. 

Radar Gain: Two-Way Transmission 


Knowing the values of 20 log A, the corresponding 
values of P 2 /P 1 are given by equation (5). Taking 
Gi = G 2 = 13.5 db, 16 tt/9 = 5.58, X = 1.5, 

10 log — = 27 + 7.5 + 17 + 40 log A- 20 log 1.5, 

Pi 

= + 40 log A -f- 48. 

56.4 Type III. Radio Gain Versus 
Distance for Given Antenna Heights 

A radar used over the sea has a wavelength of 
1.5 meters. The transmitter is 30 meters above 
sea level and the target is at an altitude of 1,000 
meters above the sea. The power gain of the trans¬ 
mitting antenna is 13.5 db, the polarization hori¬ 
zontal. The one-way radio gain is to be found as a 
function of distance. Also, the radar gain at the 
radar set by echo from the target of cross section 
a = 10 square meters is to be calculated. 


Radio Gain: One-Way Transmission (see Figure 26). 

1. The number r from Figure 15 in Chapter 6 or 
equation (115) is found to be 9.403. 

2 . u — h 2 /hi — 33.3. 

3. For r > 2 and n = 1, p is approximately equal 
to 1/r. Hence we start with p = 0.1. 

4. Equation (121), with p = 0.1 and u = 33.3, 
gives v — 2.76. 

5. From equation (112), R = 9.445. 

6. n = R/r = 1.005. Hence the target is on the 
first (i.e., lowest) lobe. Moving in the direction of 
increasing distance, the target soon approaches the 
first maximum. To get points beyond, i.e., n < 1, 
we need greater values of p but it is not necessary or 
desirable to go below about n = 0.8, since the curve 
beyond n = 0.8 is generally in the transition region 
in which the curve is more easily and more accurately 
obtained by joining the optical and diffractive curves. 
The diffractive part of the calculation is given in 
Section 5.7. Accordingly, in Table 2, the values of p 
are taken only slightly above p = 0.1 (correspond¬ 
ing to n = 1) and are diminished to find points at 
the nearer distances. 

7. To find A, we need first the free-space value A 0 , 
which is given in Figure 3 in Chapter 2 as a function 
of d. Since d T = 4.12 V30 = 22.6 km, the value of 
d corresponding to v = 2.76 is d = vd T = 62.3 km, 
and 20 log A 0 = — 111. 

8. To find the value of 

10 1o 8tJ(1 - Z>) 2 + 4Dsin%, 

A 

we need D. Since n is practically unity, correspond¬ 
ing to the first maximum, sin 2 (0/2) may be taken as 
unity. Hence the radical reduces to 1 + D. Calcula¬ 
tion using equation (103) and Figure 17 gives 
D = 0.98, and hence 

1 + D - 1.98, 

20 log (1 + D) = 6. 

Since the transmitting antenna gain Gi is 13.5 db, 
and assuming the receiving antenna gain to be 0 db, 
10 log (P 2 /P 1 ) has the value 20 log A + 13.5 or 

, P 2 

10 log 

= 4o 2 yj (1 - D) 2 + 4 D sin 2 1 GiG 2 + 0.0 db 

— 111 db + 6 db + 13.5 db 

= - 105 + 13.5 = - 91.5 db. 

























































86 


CALCULATION OF RADIO GAIN 



Figure 26. Radio gain (in db) versus distance. 













































CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 87 


Table 2. 

10 log Gi 

= 13.5 db 

; 10 log (t 2 

= 0.0db; X = 

1.5 meters; hi = 30 meters; ho = 

1,000 meters; 

cr = 10 m 2 . 

V 

V 

d(km) 

R 

n 

D 

2 

* 

20 log A 0 

20 log A 

Radio Gain 
Pi 

10 log 

Radar Gain 

Pi 

10 log pT 

0.12 

3.098 

70.49 

7.78 

0.83 

0.97 

1.30 

5.6 

-112 

-106.4 

-93 

-172 

0.11 

2.934 

66.75 

8.54 

0.91 

0.98 

1.43 

6 

-111 

-105 

-92.5 

-170 

0.10 

2.755 

62.68 

9.45 

1.005 

0.98 

1.58 

6 

-110 

-104 

-91.5 

-169 

0.09 

2.561 

58.25 

10.55 

1.22 

0.98 

1.76 

6 

-110 

-104 

-91.5 

-169 

0.08 

2.349 

53.45 

11.92 

1.27 

0.99 

1.99 

5.2 

-109 

-104 

-90 

-167 

0.07 

2.119 

48.21 

13.68 

1.45 

0.99 

2.29 

3.5 

-109 

-105.5 

-92 

-170 

0.06 

1.870 

42.54 

16.02 

1.70 

0.99 

2.68 

-1 

-107 

-108 

-94.5 

-175 

0.055 

1.738 

39.22 

17.50 

1.86 

0.99 

2.92 

-7.5 

-106 

-113.5 

-100 

-186 

0.05 

1.600 

36.4 

19.28 

2.05 

1.00 

3.22 

-16 

-106 

-122 

-108.5 

-203 

0.045 

1.457 

32.89 

21.45 

2.28 

1.00 

3.58 

-1.5 

-105 

-106.5 

-93 

-172 

0.04 

1.311 

29.59 

24.16 

2.57 

1.00 

4.04 

+3.8 

-104 

-100 

-87 

-159 

0.035 

1.158 

26.14 

27.64 

2.94 

1.00 

4.62 

6 

-103 

- 97 

-79.5 

-153 

0.03 

1.002 

22.62 

32.28 

3.43 

1.00 

5.39 

+2 

-102 

-100 

-86.5 

-159 

0.025 

0.841 

18.98 

38.67 

4.12 

1.00 

6.48 

-8 

-100 

-108 

-94.5 

-165 

0.02 

0.678 

15.30 

48.49 

5.16 

1.00 

8.10 

5.7 

- 98 

-104 

-91.5 

-167 

0.01 

0.345 

7.79 

97.08 

10.33 

1.00 

16.22 

0 

- 92 

- 92 

-78.5 

-143 


* Lobe pattern factor = 20 log f/(1— D ) 2 + 4 D sin 2 (12/2). 

Repeating the process for other values of p, a 
table of 10 log (P 2 /Pi) versus d is obtained. These 
points have been plotted in Figure 26, together with 
the points in the diffraction region obtained with the 
same data in Section 5.7.3. For sketching in the 
optical part, the value of n is kept in mind, since 
this indicates on which lobe and where on the lobe 
the point lies. 

Radar Gain: Two-Way Transmission 

To change from 20 log A = — 105 to 10 log (P 2 /Pi) 
at the radar receiver, using equation (5) and G\ = G 2 
= 13.5 db, 

10 log (P 2 /Pi) = 27 + 7.5 + 10 log o- - 20 log 1.5 
+ 40 log A, 

= 27 + 7.5 +10 - 3.5+ 40 log A, 

= 41 + 40 log A = - 169 db. 

5 6 5 Type IV. Determination of Con¬ 
tours Along which the Gain Factor A 
Has a Given Value, the Transmitter 
Height and Wavelength Being Given 

This is the so-called coverage problem which is 
treated in greater detail in Chapter 6. While the 
usual coverage diagram is derived on the basis of 
one-way or communication formulas, the diagram is 
still useful for radar since a target will return more or 
less energy to the receiver according to its position 
on the coverage diagram. The contour gain factor A 


is readily converted to P 2 /P\ for either one-waj’ 
or radar by means of equations (3) and (5). 

The method described here is more accurate than 
the graphical methods given in Chapter 6. It is best 
suited for finding maximum lobe ranges correspond¬ 
ing to a given radar gain. If A is given, and h 2 for a 
given distance d is wanted, a curve can be drawn as 
in Section 5.6.3 and then h 2 found for the given 
value of A. 

Problems 

A radar set operating over the sea has a trans¬ 
mitter with antenna height of 30 meters and a wave¬ 
length of 1.5 meters. As in the previous problems, a 
receiver with an antenna of 0 db gain is assumed in 
place of a target and the gain of the transmitter is 
again assumed as 13.5 db. The polarization is hori¬ 
zontal. The gain factor A, for illustration, is chosen 
as —130 db. Positions of the receiver are to be 
found at vdiich the gain factor takes that value. 

In Chapter 6, purely graphical methods of de¬ 
termining the contours (lobes) are given. Here we 
are concerned with finding individual points on the 
contour. Thus, for example, n = 1 if the tip of the 
lowest contour is wanted (as in range determination). 
Points near the tip require values of n near 1, such as 
n = 1.2 or n = 0.9. For the next higher lobe, the 
tip of the lobe corresponds to n = 3 and so on. The 
nulls are at n = 0, 2, 4, . . . . However, while the 
optical formula gives a null at n = 0, that is, near 
the line of sight, the true value of the field, or of A, 








88 


CALCULATION OF RADIO GAIN 


is greater. To obtain the correct result, the contours 
for the diffraction field (Section 5.7.3) and those 
obtained for the optical region should be merged 
into smooth overall curves. 

For values of r > 3, R = nr > 3, the tip and upper 
part of the lowest lobe, and the higher lobes, by 
equation (126), have a value of D close to 1. Con¬ 
sequently, equation (110) reduces to 

A = 2A 0 sin —. (132) 

2 

If n is given, sin 12/2 is determined and the calcula¬ 
tion can be performed as a free-space calculation. 
In terms of d, the equivalent free-space distance 
do , corresponding to A is given by Figure 3 in Chapter 
2 and is equal to 

d = 2do sin —. (133) 

2 

In the stated problem r > 3, so that the free-space 
calculation suffices. This is given in paragraph (1). 
In paragraph (2), the method including the diver¬ 
gence is given. 

1. Free space. In the given problem, r = 9.4, so 
that the simplified calculations should be sufficient. 
The value of d 0 corresponding to 20 log A — —130 
is found from Figure 3 in Chapter 2 to be 566 km. 
Hence by equation (133) for n — 1, the true distance 
d for complete reinforcement by the reflected wave 
is twice as much and 

d = 2.(566) = 1,132 km. 

For n = 1.2,12/2 = 0.67r [from equation (116)], giving 
d = 2.(566)sin(0.67r), 

= 1,076 km. 

To find the corresponding height h 2 , a curve of the 
type in Section 5.6.3 is needed for —20 log A versus 
height. From this, the height corresponding to 
20 log A = —130 can be read. Alternatively, the 
calculation given later in (2) will determine both 
d and h 2 . 

If instead of A, either the radio gain or radar gain 
is given, A is first found from equations (3) or (5). 

2. Calculation including divergence, n = 1. As in 
the previous problem, r = 9.4 (from Figure 15 in 
Chapter 6). A convenient value of p, approximately 
equal to 1/r, is selected. In this case, we take 
p = 0.1. This value is then improved by applying 

Newton’s method pi = p — —— to the equation 

f'(p) 

f(p ) = - - - \l (1 - Df + 4Z> sin 2 -, (134) 

Vo v y 2 


remembering that D is a function of p as given in 
equation (117), where v Q is the value of v which corre¬ 
sponds to the distance do at which the given value of 
A would occur in free space. 

If n = 1, equation (134) reduces to 

- = -(! + £»)• (135) 

^0 V 

The correction formula corresponding to equation 
(134), denoting by pi the improved value of p, is 


v — Vo 


P 2 D 2+ 

The correction formula corresponding to equation 
(135) (n = 1) is 

1 (1 + D) 


\f 


(1 - D ) 2 + 4 D sin 2 (Q/2) 


3(1 - D 2 ) 
4 pD 


[(i-w+^XS) 


p J 

(136) 


pi = p- 


Vo V 


1 + D + 3.D / 1 + P 2 \ 
p 2 D 2 2 pv \ 1 — p 2 ) 


(1 - D 2 ) 


(137) 


Taking n = 1, d 0 from A = 3\/St d Q (or Figure 3 
in Chapter 2) is 566 km, and 


566 

dx 


566 

22.5 


= 25.1. 


Taking p = 0.1 as an initial value as explained above, 
then D is found to be 0.9788 [equation (117)]. 
Using — nr — 9.4 and equation (113), 
v = 165.6. 

Substituting these values in equation (135) gives 
pi = 0.10385. 

Repetition of this procedure requires no change in 
these five figures. Accordingly, for n = 1 and 
A = —130 db, p = 0.10385. The corresponding 
values of D and v are 

D = 0.9789 

v = 49.38, d = 1,110 km. 

In (1) d was found to be 1,132 km. Now s = p/v 
= 0.0021. From equation (121), 

u= — = 2,898 

h 

h 2 = 86,940 meters. 


For n = 1.2, the calculation proceeds as for n — 1 
except that equation (134) rather than equation (135) 
is used: 


R = 11.283. 












CALCULATIONS FOR OPTICAL-INTERFERENCE REGION 


89 


The value of p by successive approximations in 
equation (136) is found to be 
p = 0.08127 
v = 47.35, d = 1,065 km 
D = 0.9851 
s = 0.00184 

- = 2,772 

K 

h 2 = 83,160 meters 

3. Vertical polarization, p + 1 , 0 + 180° (sea 
water). The procedure given here is first to find 
the position of the point for a given n assuming the 
reflection coefficient to equal — 1 and then find the 
shift caused by the change in the reflection coeffi¬ 
cient. For the most important case, n = 1 , the new 
maximum distance is given by 

d = do (1 + pD), (138) 

where d 0 is the free-space distance corresponding to 
the given value of A. p is found from the value of 0, 
the grazing angle at the reflection point given by p 
found above in the calculation for p = 1 and 
0 = 180°. For the new contour tip p changes, but 
the angle \f/ does not change sharply, so that p and 0 
as found for the p obtained by the simplified calcula¬ 
tion is a close approximation. 

For p^ 1 , 0 7 ^ 180°, £2 = 5 + 0 — 7 r [see equa¬ 
tion (29)] consists of two parts. 8 = irR/r = (A/X) 2t 
= rnr and 0' = 0 — tt. Writing 12 = Ntt, N the lobe 
number for the case p ^ 1 , 0 ^ 180°, the require¬ 
ment for the new lobe tip is N = 1 ; for the first 
maximum 12 = 7 r = Rir/r + 0 — 7 r where R is the 
path difference variable at the new lobe tip. Hence 
the value of R at the new tip is given by 

R = r (2 — 0/t r). (139) 

Using the results in (2), 

p = 0.10385, 

D = 0.9788, 
r = 9.4, 
d 0 — 566, 

we find from Figure 24 or equation (107), 0 = 0.725°; 
from Figures 14 and 15 in Chapter 4, p = 0.76, 
0 = 170°. Substituting in equation (138), 
d = 566 (1.744) = 987 km. 

From equation (139), 

R = (9.4) (1.056) = 9.92. 

Also 

d 987 , on 
d^ 22.5 


The above value of p can now be improved by 
substitution in equation (137) with D replaced by 
pD which gives 

p = 0.0985. 

In the calculation just made, p has been assumed 
constant. This can be checked by finding 0 as de¬ 
termined by the new value of p. The result is 
0 = 0.766° and the corresponding values are p = 0.74 
(as against 0.76 previously) and 0 = 169° (as against 
170°), which is good enough. We now find 


s 


p = 0.0985 
v ~ 43.9 


0.00224 


- = 2,359 

K 

h» = 70,770 meters. 

Hence the new maximum point of the lobe is at a 
distance of 987 km and at an elevation of 70,700 
meters, as compared with a distance of 1,110 km 
and height 86,900 meters for perfect reflection. 


5 6 6 Maximum Range Versus Receiver 
(or Target) Height 

If the value of A has been determined by using 
the minimum detectable power in equation (3) 
or (5), the corresponding contour is a curve of 
maximum range versus receiver height for com¬ 
munication or maximum range versus target height 
for radar. Generally, the lower part of the lowest 
curve (n < 1) is of greatest interest. If A is suffi¬ 
ciently small (i.e., 20 log A numerically large and 
negative), the complete contour has points below 
the line of sight. If the transmitter antenna is low 
(hi < 30\ 2/3 meters), the lower points are likewise 
given by the diffraction formula, discussed in Sec¬ 
tions 5.1.7 and 5.7.3. Several such curves, the lower 
part of the lowest lobe corresponding to various 
transmitter heights for X =1 meter and 20 log A 
= — 130, are given in Figure 27. 

Consider, for example, the curve for hi = 2 meters. 
The uppermost points of the curve correspond to 
the tip of the lowest lobe and were found by the 
procedure used in Section 5.6.5, putting n — 1. 
The lowest points were found from the diffraction 
formula by the method of Section 5.7.3 with the 
aid of Figures 31 to 36. It has been pointed out 
that for n < 1, the optical interference formula is 
inadequate. 

To locate a point between the upper extreme 
(n = 1) and the diffraction points, a curve of A 




90 


CALCULATION OF RADIO GAIN 



Figure 27. Maximum range versus receiver height h 2 for given values of transmitter height h u 


IN METERS 
















































































































BELOW THE INTERFERENCE REGION 


91 


versus h 2 for some distance is constructed. In 
Figure 28, a set of such curves is given for the dis¬ 
tance 100 km and for various transmitter heights. 
By taking the intersection of 20 log A equal to —130 



Figure 28. Radio gain versus receiver height X 2 , for 
given values of transmitter height hj. 


with the curve hi = 2, a value of h 2 is obtained. 
This value h 2 = 3,300 and d = 100 km represent 
the coordinates of a point on the contour hi = 2 in 
Figure 27. 

It is of some interest to observe the shortening 
of the lobe for hi = 100 meters on account of the 
divergence winch is close to unity at the tips of the 
lobes corresponding to the low transmitter heights 
but drops to 0.65 at the tip of the lobe for hi = 100. 
Since for hi = 100, the formula for both antennas 
low (h < 30X 2/s ) is not applicable, the lowest points 
on the curve hi = 100 in Figure 27 were obtained 
by applying the reciprocity principle to the curves 
already obtained. The distance at which h 2 = 100 
on the hi = 2 curve is the same as that at which 
h 2 = 2 on the hi = 100 curve. 


5 7 BELOW THE INTERFERENCE REGION 
571 Analysis of the First Mode 


significant factors, one for free space and one for 
the earth effect. The latter may be divided into a 
plane earth factor and a shadow factor. 

1. Free space. It has been shown in equation (18) 
in Chapter 2 that for doublet antennas, with matched 
load at the receiver and adjusted for maximum 
power transfer, the free-space gain factor A 0 is 
given by 



For other types of antennas in free space, this takes 
the form 

4 0 VG)V(4 =J~ = — V~^V,77. (140) 

' P, 8 ml 


Under actual conditions vdien earth and atmos¬ 
pheric effects are of importance, each mode will be 
considered to have A 0 as one factor. 

It may also be recalled that for given pov T er P 2 
delivered to the load, the corresponding electric- 
field strength, under matched conditions, is given by 


E 


8W5 

X 



(141) 


Combining equations (140) and (141) gives the 
free-space value of the electric field strength E, in 
terms of transmitted powder 

E = VpW(7;, (142) 

d 


w r hich is the same as equation (7) in Chapter 2 with 
the addition of the transmitter gain G\. 

2. Plane earth. The earth modifies the field by 
absorbing and reflecting radiation. If the earth 
were plane and perfectly conducting, the value of 
the gain factor w’ould be 24 0 and the electric field 
2 Eq for vertical antennas several wavelengths above 
the ground and at distances sufficiently large. The 
imperfect conductivity of the earth produces a 
change in the gain. Representing this effect by the 
factor 4.i (where Ai < 1), 


Except for the numerical constants involved, the 
discussion for one mode applies to all the other modes. 
Each mode is of the form 4>(d) • f(hi) • f(h 2 ), i.e., the 
product of a distance function $ by two antenna 
height-gain functions / (see Section 5.1.7). 

Distance 

3>i(< 2), the distance function of the first mode, can 
be represented as the product of tw r o physically 


A = 2A 0 Ai. (143) 

The plane earth factor Ai depends on distance and 
on the electrical properties of the earth. Fortunately 
the earth constants enter the problem in an intrinsi¬ 
cally simple way, as the main effect is taken into 
account by multiplying the distance d by a certain 
factor which we shall denote by p', so that Ai is 
mainly a function of p'd. The new parameter p' is 




















































92 


CALCULATION OF RADIO GAIN 


different for the two states of polarization. For 
vertically polarized radiation, 


27r e r 


X Nc | 2 

where e c is the complex dielectric constant e r — j’60crX, 


or 


2ir V(e r - 1) 2 + (60<rX) 2 


X v + (60<rX) 2 
For horizontal polarization, 


(144) 


V' = — U ~ 1 I = — V( t r - l) 2 + (60crX) 2 . (145) 
X X 

Ai depends also on the phase of the complex di¬ 
electric constant. The phase is determined by the 
parameter 



For ultra-short waves, with the exception of 
vertically polarized waves over sea water at distances 
less than 50 /p' (see Section 5.7.4 and Figure 45) and 
a wavelength greater than 1 meter, A\ is inde¬ 
pendent of Q and, to a sufficient approximation, is 
given by 

.4, = -L • (147) 

p'd 

A \ as a function of p'd is plotted in Figure 56. 

In the case excepted above, Ai deviates substan¬ 
tially from l/p'd (see Figure 45) for distances less 
than 50/p'. Table 3 gives 50/p' as a function of X. 
It appears that the deviations are immaterial for 
practical purposes as long as the wavelength is 
smaller than 3 meters, since we are usually con¬ 
cerned with ranges larger than 7 km. It should 
further be mentioned that, in the above case, Ai 
depends to a small degree on Q. However, the 
variations are less than 1 db and may be neglected 
for wavelengths less than 10 meters. 


Table 3. Sea water (vertical polarization). 


X 

1 2 

3 

4 

5 

6 

7 

8 

9 

10 

meters 

50 

p' 

2 4 

7 

12 

17 

24 

30 

50 

71 

200 

km 


The condition that the earth may be considered 
plane is that the shadow factor F s [see equations 
(149) and (207)] shall be approximately unity. For 
F a — 0.9, which is approximately 1 db below unity, 
7i = tf{8) = 0.4 from Figure 58; f(8) in Figure 57 


is approximately unity, so that = sd = 0.4. 
Since s, from equation (150) for k = 4/3, is given by 
4.43 X 10“ 5 X“ 1/3 , it follows that d, for the plane-earth 
approximation to hold, must be less than 10 4 X 1/3 , 

d < 10 4 X 1/3 (plane earth). (147a) 

3. Curved earth. The screening effect of the earth 
curvature results in a further decrease in gain. Well 
within the diffraction region, the shadow effect pro¬ 
duces an exponential drop in field strength with dis¬ 
tance, which is much greater for higher modes than 
it is for the first mode. 

Denoting the screening or shadow factor by F s 
and the distance gain factor by $i, we have 

*i = 2 AqA\F 8 . (148) 

For the dielectric case (see Sections 5.1.7 and 5.7.3) 
8 > > 1 and for distances greater than 1.5/s, the 
shadow factor is 


where 


F, = 2.507 (sd) v2 e~ 1607 (8<i) , (149) 


f 2jr 1 1 1/3 

L X (kay-j ’ 

= 4.43. 10-V^(±V 2/3 
\3 kj 


(150) 


Equation (149) gives the value of the shadow 
factor for the first mode only. In Figure 32, the 
curve marked “dielectric earth” is a plot of the 
shadow factor evaluated by using all terms or modes. 
However for sd > 1.5 only the first mode is impor¬ 
tant. Consequently, equation (149) represents this 
curve accurately for all values of sd larger than 1.5. 


Height-Gain 

For antennas at zero height, the height-gain 
functions/ are equal to unity, so that equation (148) 
represents the actual value of the first mode for 
both antennas at zero height. 

When the antennas are raised above the ground, 
it is convenient to distinguish between low and high 
antennas, the division between the two cases being 
given by the critical height h c = 30X 2/3 . 

1. Low antenna ; h < h c = 30X 2/3 . For a low 
antenna, f is a function of Ih and of Q, where l is a 
quantity that depends on the complex dielectric 
constant and is given by 



(151) 












BELOW THE INTERFERENCE REGION 


93 


in which p' is the distance coefficient given by 
equations (144) and (145). Let the value of / for 
a low antenna be denoted by H L . 

The magnitude of H h is given by 

HL = ^ 1+ WTi + m - (152) 

and is plotted in Figure 47. 

For height h larger than 4/1, the two first terms 
under the square root may be neglected in comparison 
to the third term, and H L becomes approximately 

II L ^ Ih. (153) 

In order to show more clearly when this approxima¬ 
tion is justified, Table 4 gives values of 4/1 for differ¬ 
ent wavelengths and vertical polarization. For 
horizontal polarization, 4/1 is quite small. 



Figure 29. Height-gain as a function of height. (See 
Figures 7, 25, and 47.) 

Inspection of Table 4 shows that, except for the 
case of sea water at wavelengths above 1 meter, 
the approximate equation (153) is good for heights 
above about 50 meters. 


Table 4. Values of 4 Jl for different wavelengths 
(vertical polarization). 


A in meters 

0.1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Sea water 

0.06 

10 

28 

50 

80 

115 

133 

160 

235 

285 

300 

Fresh water 


6 

11 

17 

22 

30 

33 

40 

44 

53 

57 

Moist soil 


3 

7 

11 

15 

18 

21 

25 

28 

33 

36 

Fertile ground 


3 

5 

8 

10 

13 

16 

17 

20 

25 

27 

Very dry ground 


2 

3 

4 

6 

8 

9 

10 

12 

15 

16 


To a first approximation, the value of / for low 
antennas is the same for all modes, so that the height- 
gain functions may be factored out, as was pointed 


out in Section 5.1.7. The gain factor for the case 
when both antennas are low can then be represented 

by 

A = 2 A 0 A 1 F a (H l \(H l ) 2 , (154) 

where F s is sum of the shadow factors of all the 

modes. F s has been plotted in Figure 32. Equation 
(154) is also valid in the optical region, provided 

d >> —. (155) 

X 

This condition is added to insure that the point is 
well below the center of the lowest lobe. 

2. Elevated antenna ; h > 30A 2/3 . In this case the 
height-gain function / increases exponentially. Rep¬ 
resenting the increase over Ih by g, we have 

/ = glh. (156) 

For the dielectric case 8 > > 1, refer to equation 
(193) to define 8. (See Sections 5.1.6 and 5.7.3.) 

g = 0.1356 (eft)' 0 - 904 10°- 948 ^ 

where 

4^y /3 

AW ’ 

=!x - 2 ' 3 (±y s . 

60 \3 kJ 

See Figures 35 and 36. 

For the case of sea water, vertical polarization and 
wavelength in the VHF (1 to 10 meters) range, 
g requires a correction factor g' which depends on A. 
A table of g f is given with the graph of g in Figure 36. 

The change of / with height h is represented 
schematically in Figure 29 (see also Figures 7, 25, 
and 47). 

Formula for the Dielectric Earth. (See Sections 
5.1.7 and 5.7.3.) 8 > > 1. 

The first mode has the form 4>i (d) • f(h i) • f{h 2 ). 
If the value of the gain is given substantially by the 
first mode, and substitution from equation (148) is 
made, it is found that 

A = [2AoAiF s ilf(hi) • f%). (160) 

For the dielectric case equation (160), using equa¬ 
tions (140), (147), (151), and (156), becomes 

A=l^{gh) l {gh\. (161) 

2 dr 

The same formula holds for sea water, vertical 
polarization, wavelengths in VHF range, d > 50/p', 



(157) 

(158) 

(159) 



















94 


CALCULATION OF RADIO GAIN 


and h > 4/1, as described above, except that F s or 
F sl is represented by various curves in Figure 32, 
according to the value of X, and g is modified slightly 
by a correction factor g'. 

Formula for Sea Water. Vertical Polariza¬ 
tion. VHF. 

Equation (160), the general formula for the first 
mode, in the VHF (1 to 10 meters) range, becomes 

A = [2A 0 Ai F 8l ](H L gg')i(H L gg') 2 , (162) 

where H L is the low antenna height-gain function 
whose formula is given by equation (152), and 
gg' = 1 for low antennas. As pointed out above, 
if h > 4/1 and d > 50/p', equation (162) reduces to 
equation (161). g' , the correction factor for g, is 
given in Figure 36. For a more extended discussion, 
see Section 5.7.4. 

5 7 2 Effect of Changing the Value of k 

In the optical region, the effect of a linear gradient 
of refractive index has been shown to be equivalent 
to replacing the radius of the earth a by an effective 
radius ka and then treating the atmosphere as 
homogeneous (see Chapter 4). 

In view of the equivalence of the sum of modes to 
the optical formula in the optical region, it follows 
that the modes should be changed in the same way 
in the optical region, i.e., a should be replaced by ka. 
These same modes supply the solution of the wave 
equation in the diffraction region, so that in both 
regions the substitution of ka for a will take care of 
an atmosphere with a linear variation of refractive 
index with height for all values of k. 

For given transmitter and receiver antenna heighis, 
hi and h 2 , the first maximum of the field-strength 
versus distance curve (see Figure 4) will frequently 
represent the limit of detection. The first maximum 
occurs not far from the line of sight which, for a 
standard atmosphere (k = 4/3), is given by 

( Il — ^ 2 a (V/ix + V/i 2 ). (163) 

If 4/3 is changed to k, the more general form is 

d L ' = y/2ka (V/ix -f- V/* 2 ). (164) 

The first maximum will now be near 



Suppose next that the point at distance d L ' for the 
given heights was originally well within the diffrac¬ 
tive region. The exponential part of the gain due to 
change of k from 4/3 is equal to 

- 1.607s ( d L '-d L ) 

or 

e -1.607 W3k/4-l)sd L ' 

The above formula is obtained by combining equa¬ 
tion (149) for F s , and s is obtained from Figure 31 
(k = 4/3). This corresponding gain in decibels is 
20 logxo g-i-607 (V3*74 -1 )sd L 

= 20 1.607 X .434 j - •) , 

= - 14 l) db, 

below the free-space value. Since a maximum is now 
near d L f as a result of changing 4/3 to k, i.e., the 
field has a value of 6 db above the free-space value, 
the gain is approximately the exponential value 

- 14«fe(-Jj-l), 

as a consequence of refraction giving a value k 
greater than 4/3. For k — 12, and sd L = 5, the gain 
would be about 140 db at some point near d L ' = 3 d L , 
at heights hi and h 2 such that 

d L ' — V2 ka (V/q + V/i 2 ) 

and such that h 2 is well within the diffraction region. 

For given transmitter height hi and distance d, the 
effect of increasing k is to lower the lowest lobe 
roughly by the amount by which the line-of-sight 
elevation at the distance d is lowered in changing 
from 4a/3 to ka. At this distance the height of the 
line of sight is h L , and 

V/j7 = ~L= - V// (for k = 4/3) (165) 

becomes 

^ - vS " v * 7 ' (166) 

so that there is a downward shift of approximately 

hL ~ hL> = !r 0 “ I) - 2 ^l d ( l “ >ls) 

(167) 






BELOW THE INTERFERENCE REGION 


95 


If hi is small compared with h L and h L f , the result is 
approximately 



Receivers situated between h L and h L ' were formerly 
below the line of sight but are now above it (assum¬ 
ing k > 4/3), with a consequent substantial gain in 
signal strength at some points and loss at others. 
The chances of detection in the region, however, have 
improved. 

If both antennas are low , a change of 4a/3 to ka 
gives a field strength change such that the field 


where 



where k may now have any value, 
change to A r where 



(169) 

(170) 
A itself will 


(171) 


Figure 30 illustrates the values of (3k/4) n for various 
values of k and n used in equations (169), (170), and 



O.l 0.5 I 5 10 50 100 500 1000 5000 10,000 

Figure 30. Values of (3 kj 4) M . Note: Values shown between 0.1 and 0.5 of the ordinate scale are minus values. 


which was formerly at a point d well within the 
diffraction region will now be found at a distance of 
approximately 



except for the decrease in free-space gain (20/3) log 
3k/4, occasioned by the increase in distance. If, for 
instance, k = 12, the free-space gain is equal 
approximately to —7 db, and the ratio of the new 
distance to the old is equal to (3fc/4) 2/3 == 9 2/3 == 4. 

More generally, if A/A 0 is known at a point 
( h 2 ,d ) for a transmitter height hi and for k = 4/3, 
the same value of A/Aq will be found at W, /? 2 ', d 


(171). Figure 43 gives the same information in 
nomographic form. 

Hence if a coverage diagram is known for k = 4/3, 
then the same diagram can be used for k 9^ 4/3 if the 
diagram is interpreted in terms of h', d', and A'. 

5 7 3 Graphs for the Case 

of the Dielectric Earth (S>>1) 

1. Fundamental formula for gain factor. This formula 
is (see Sections 5.1.7 and 5.7.1) 

A =lj_(gh)i(gh) i , (172) 

where g = 1 when h < 30X 2/3 . 



















































































96 


CALCULATION OF RADIO GAIN 



10 5 3 2 i 0.3 0.1 .0 3 .01 

WAVELENGTH IN METERS 


Figure 31. sf(8) versus wavelength and frequency. For dielectric earth and k — 4/3, f(8) = 1, and s ■= 4.43 X lO^X" 1 / 3 . 
(See Figure 57 for/(5).) 










































































BELOW THE INTERFERENCE REGION 


97 



Figure 32. Shadow factor Fs, 20 log F s (evaluating all modes), or 20 log F s /(sd) 2 versus ^sd. Curve for dielectric earth, 
20 log Fg /( sd) 2 , is the same as curve 0 in Figure 58. 





























































20 LOG X 


98 


CALCULATION OF RADIO GAIN 



Figure 33. Shadow factor for dielectric earth. 



Figure 34. 20 log x versus x. 
































































BELOW THE INTERFERENCE REGION 99 


FREQUENCY IN MEGACYCLES 





Figure 36. g and ^ versus ehg (5) 

















































































































































100 


CALCULATION OF RADIO GAIN 



Figure 37. ehz = eu versus sd = sp. Points eh,sd, lying to right of dotted line corresponding to ehj for a given trans¬ 
mitter indicate that the points lie in the diffraction region of the transmitter, or d > d^. Diffraction region dielectric earth 
— curve parameter: 20 log A = 20 log A — 20 log h x — 20 log g x . 


eh = eu 


































































O CD 


BELOW THE INTERFERENCE REGION 


101 



Figure 38. eh 2 = eu versus sd = sv. 


BOTH ANTENNAS LOW 


h V 3 °x- 


,2/3 


20iog a= 20 log a- 20 log h, 



Figure 39. eh 2 = eu versus sd = sv. 













































































DIELECTRIC EARTH 


102 


CALCULATION OF RADIO GAIN 



sd 

2 r— 


d (km) 

IOf— 


i The d and sd scales may 

fc be multiplied by any 

jp suitable power of 10 to 

j- accommodate values of d 

E greater than 10 km 


.4 



4 





i Relation of X, sd and d 


.02 


I 


Figure 40. Relation of X, sd [representing sdf(8)], and d. See Figure 31. f(8) = 1 for 8 > > 1 











BELOW THE INTERFERENCE REGION 


103 


X(m) 


x 

i- 

ct 

< 

u 


tr 

K 

u 

u 

_l 

u 


.01 


.02 

.0 3 

.04 

.05 


.1 


— r —.01 



-.04 


.05 




2— 


3 ~ 


3 


4 —= 


5 


5 



eh 20 LOG g 





h(m) 20 log h 

1000—3—60 



Figure 41. eh [representing ehgib)], 20 log g and g versus X. See Figures 35 and 42, 









































104 


CALCULATION OF RADIO GAIN 


For the dielectric earth, 5 > > 1. See equation ^193). 
If both antennas are low (h < 30X 2 3 ) equation (172) 
and the accompanying figures (Figures 31 to 41) are 
valid for all distances d such that 

d>> —. (173) 

X 

If one or both antennas are elevated, equation (172) 
is valid only well within the diffraction region of the 
transmitter, i.e., for 

d>>d L . (174) 

The following quantities required to find 20 log A 
are given in Figures 31 to 41: 

$ as a function of X is given in Figure 31. 

20 log F# versus srf in Figures 32 and 33. 

20 log d can be found by using Figure 34. 

( as a function of X by Figure 35. 

20 log g versus eh i s given by Figures 36 and 41. 

When one antenna is low, h < 30X 2 3 , and the other 
quite elcixited, h > 1,200X 2 3 , a result valid for near 
the line of sight can be found from the formula and 
graphs in Section 5.7.5. obtained by summing several 
modes. 

A more general method of finding the gain near 
the line of sight is to use equation (172) well below 
the line of sight to obtain a curve of A versus h* and 
by constructing a similar curve for the optical region 


V C » 109 g ti 20 log h 



Figure 42. Illustrating use of Figure 41. Note: (eh 
represents ehg(6i): iefc* = 12.3] represents [eh^g(8) = 
\2J$l 


by the method of Chapter 6, “Coverage Diagrams.” 
By joining the two curves into a smooth overall 
curve, it is possible to estimate A in the transition 
region near the line of sight. 

For the case of short distances and receiver below 
the interference region, see Section 5.1.7. 

2. For h < 4/1. Vertical polarization. A more 
accurate result can be obtained by replacing the 
height-gain ( gh ) by H L /l or in decibels by 20 log H L 
— 20 log l. H l is given by Figure 47 and l by 
Figure 46 (see Table 3). 

3. Graphical aids ( continued ). 

A. Definition of A. Figures 31 to 36 can be com¬ 
bined into a form more convenient for numerical 
computation. In Figure 37, a curve parameter A is 
introduced, defined by 

A = A , (175) 

hiffi 

where gi is a function of ehi. This may also be ex¬ 
pressed in the form 

20 log A = 20 log A — 201og%i. (176) 

(For hi < 4//, 20 log A = 20 log A — 20 log H L -j- 20 
log l.) 

Equation (172) can be written as 



A = 1.77 X 10' 7 — 

W 

(177) 

or 


F. 

(177a) 

20 log A = 

where 

— 135 + 20 log -~r + 20 log^, 

(sdy 

<P = (.ehi)gt, 


with g 2 a function of eh 2 . Note that F t /(sd) 2 is a 
function of sd only and is independent of height. 
While hi usually represents the transmitter antenna 
height and h 2 that of the receiver, the role of hi and 
h 2 in equations (176) and (177) may be interchanged. 

To facilitate the use of Figure 37, three nomograms 
have been added (Figure 40 gives sd when X and d 
are given. Figure 41 gives eh h 20 log h and 20 log g 
when X and h are known. Figure 43 gives the modi¬ 
fied height h! and distance d' for given h, d, and k.) 
To find sd for a value of d which is not on the nomo¬ 
gram, say 120 km, find sd corresponding to a distance 
100 times smaller (i.e., 1.2 km) and multiply result¬ 
ing sd by 100. Proceed similarly for eh. 

B. Both antennas low. In the case of both an¬ 
tennas low’, hi and h 2 < 30 X 2/3 , the contours 20 log A 








BELOW THE INTERFERENCE REGION 


105 



Figt~b£^43. Relation c4 k.d to h'ji' a nmctioo of L 











106 


CALCULATION OF RADIO GAIN 


are given by Figure 39. If both antennas are so low, 
say, hi and h 2 < 4/Z (see Table 4) that it is desired 
to use H l for greater accuracy for vertical polariza¬ 
tion (Figure 47), then for 20 log A we take 20 log A 
+ 20 log l — 20 log H l (see Section 5.7.3) and 
instead of eh 2 in Figure 39, we use as ordinate eH L2 /l. 
Then for given sd, Figure 39 would give the value 
of eH L2 /l . If the frequency is given, e/l is known 
(Figures 35 and 46), and we must find the value of 
lh 2 v r hich corresponds to a known value of H L2 (Fig¬ 
ure 47) for the appropriate value of Q = e r /60<jX. 
From lh 2 , li 2 is found by dividing by l. 

If only one antenna height is less than 4/Z, then de¬ 
fine 20 log A as 20 log A + 20 log l — 20 log H L for 
that antenna and h 2 then refers to the other antenna. 

C. Non-standard atmosphere, k ^ 4/3. The pre¬ 
ceding graphs are all based on k — 4/3. If k 5 * 4/3, 
h h h 2 , d, and A should be replaced by hi, h 2 , d!, and 
.4', wdiere 

»' -» 

-.(?r 

or 

/3A 2/3 

20 log A' = 20 log A + 20 log (^ J . (178) 

The change of h,d, A to the primed values can be 
made with the aid of Figure 43, i.e., if h,d are known, 
change to h',d', then Figure 37 w'ill give A', which 
in turn v r ill give A', and this with the aid of Figure 43 
will give A. 

D. Change to dimensionless coordinates. In the op¬ 
tical region, convenient coordinates are (see Section 
6.5) 

d d^ 

^2kahi d T 



For these coordinates, equation (175) becomes 


hg(e) 

Writing 

e = eh h 
s_ = s^2kahi, 


(179) 

(180) 


it follows that 


eh 2 = eu, 
sv = sd, 

and, using equations (150) and (159), 
s 2 = 2e. 

Consequently, Figure 37 can be used with sv 
replacing sd, eu replacing eh 2 , and A is defined in 
equation (179). 

Caution: In using the graphs, care must be exercised 
when one or both antennas are elevated to see that the 
receiver antenna is well within the diffraction region, 
i.e., 

d>> d L . (181) 

E. Illustrative problems; diffraction formula; di¬ 
electric earth. In Section 5.6, four types of problems 
were considered for the optical-interference region. 
The same four types are given here, for a receiver 
below the optical-interference region. A dielectric 
earth is assumed so that the figures in Section 5.7.3 
are applicable. These require supplementing by 
equations (3) and (5). 

■i 

For one-way transmission the radio gain is 

10 log — = 20 log A + 10 log ((?!<?,). (182) 

Pi 

For two-way transmission the radar gain is 

10 log — = 40 log A + 10 log ( GiG 2 ) 

Pl + 7.5 + 10 log <r - 20 log X. (183) 

Type I. The heights and distance apart of the 
transmitter and receiver antennas and the wave¬ 
length are known. The radio gain is to be found. 

An early-w r arning set has a horizontal antenna, 
located 118 meters above sea level. A receiver is 
located in an airplane 1,520 meters above sea level, 
at a distance of 300 km. The wavelength is 3 meters. 
The gain of the radar antenna is 96 db and its pov T er 
output 100 kw. (a) The power received by the air¬ 
plane receiver, assuming a gain of 10 db, is to be 
found, (b) The power returned to the radar by the 
airplane, assuming that* the airplane has a radar 
cross section <7 of 40 square meters, is to be found. 

One-way: From Figure 2, d L = 205 km. Hence 
the receiver may be assumed well within the diffrac¬ 
tion region. 

From Figure 40, sd = 9.3, with/(5) = l. 

From Figure 41, eh 2 = 12.3, with g(8) = 1. 

From Figure 37, 20 log A = — 213. 







BELOW THE INTERFERENCE REGION 


107 


To convert 20 log A to 20 log A by equation (175), 
we need 20 log hi and 20 log g h which are given 
by Figure 41: 

20 log hi = 41.3, 

20 log gi = 1.5. 

Hence 

20 log A = —170, 


and by equation (182) 


10 log— = -170 + 96+ 10 = -6+ 

Pi 

Since Pi is 10 5 watts, 

P 2 = 10 5 X 10“ 6 ' 4 = 10“ 1,4 w. 

Radar: Substituting in equation (183), the radar 
gain is given by 

10 log (P 2 /P 1 ) = -340 + 2(96) + 7.5 + 16 - 9.5, 
= —134 db 


or 

P 2 = Pi X 10" 13 - 4 . 


The power output Pi is 10 5 watts, so that the max¬ 
imum received power 


p 2 = 10" 84 w. 


The minimum detectable power of the set is given as 
1.6 X 10“ 8 = 10“ 79 watt, so that under the given 
conditions the power returned b}^ the target would 
be slightly below the threshold of detection. 

Type II. Gain versus receiver (or target) height 
is to be found for given distance, given wavelength, 
and given transmitter height: A radar has an antenna 
height of hi = 30 meters, a wavelength of X = 1.5 
meters and a distance from a receiver (or target) of 
d = 100 km. Assuming a receiver antenna gain 
G 2 = 1, the variation of P 2 /Pi at the receiver with 
receiver height is to be found. Also, assuming a 
target of cross section a = 50 sq meters, the varia¬ 
tion of P 2 /Pi at the radar receiver with target height 
is to be found. The radar antenna has a gain of 
13.5 db. 

One-way: 

sd = 3.9 (from Figure 40). 

From Figure 37, for the fixed value of sd = 3.9, we 
find a correspondence between values of 20 log A and 
eh 2 , listed in Table 5 below. Ity means of Figure 41 
or equation (159), eh 2 is changed to h 2 and by means 
of equation (176), A to A. From Figure 41, it is seen 
that 20 log hi = 29.5 and 20 log gi = 0. To change A 
to P 2 /P 1 , the transmitter gain of 13.5 db and the 
receiver gain of 0 db must be taken into account, 
according to equation (182). The result is given in 


Table 5. The values of 10 log (P 2 /P 1 ) are plotted in 
Figure 25, together with the results found with the 
same data in Section 5.6.3 for the optical-inter¬ 
ference region. 


Table 5* 


h 2 

Meters 

eh 2 

20 log A 

20 log A 

Radio 
Gain 
in db 

Radar 
Gain 
in db 

63 

0.8 

-190 

-160.5 

-147 

-273 

142 

1.8 

-180 

-150.5 

-137 

-253 

259 

3.3 

-170 

-140.5 

-127 

-233 

417 

5.3 

-160 

-124.5 

-117 

-208 


* See also Table 1 and Figure 25. 


Radar: 

10 log — = 40 log A + 27 + 7.5 + 10 logo- 
Pl - 20 log 1.5, 

= 40 log A + 27 + 7.5 + 17 - 3.5, 

= 40 log A + 48. 

Type III. Gain versus distance is to be found, 
with antenna heights and wavelength given: Using 
the same data given in Section 5.6.4, the gain as a 
function of distance in the diffraction region is to be 
found. The result has been plotted in Figure 26. 
The polarization is horizontal. 
hi = 30 meters Gi = 22.4 (13.5 db) 

h 2 = 1000 meters G 2 (one-way) = 1 (0 db) 

X = 1.5 meters G 2 (radar) = 22.4 (13.5 db) 

<j = 10 square meters 

From Figure 41, 

eh 2 = 12.5. 

Referring to Figure 37, we find a correspondence 
between A and sd. Values of A are to be assumed. 
To change sd to d, use Figure 40. To change A to A, 
use equation (176). From Figure 41, 

20 log 30 = 29.5, 

20 log gi = 0, 

20 log A = 20 log A + 29.5 + 0. 

The radio gain is then given by: 

One-way: 

10 log — = 20 log A + 13.5 + 0; 

P1 

The radar gain is then given by: 

Radar: 

10 log— = 40 log A + 7.5 + 27 + 10 - 3.5 
Pl 

= 40 log A + 41. 












108 


CALCULATION OF RADIO GAIN 


These equations are evaluated in Table 6 and the 
one-way values are plotted in Figure 26. 


Table 6 


20 log A 

sd 

d 

20 log A 

Radio Gain 
in db 

Radar Gain 
in db 

-170 

6.2 

159 

-140 

-127 

-239 

-180 

6.9 

177 

-150 

-137 

-259 

-190 

7.6 

193 

-160 

-147 

-279 

-200 

8.3 

213 

-170 

-157 

-299 

-210 

9.0 

231 

-180 

-167 

-319 

-220 

9.7 

249 

-190 

-177 

-339 

-230 

10.3 

263 

-200 

-187 

-359 

-240 

11.0 

282 

-210 

-197 

-379 

-250 

11.8 

305 

-220 

-207 

-399 

-260 

12.5 

320 

-230 

-217 

-419 

-270 

13.2 

340 

-240 

-227 

-439 

-280 

13.8 

357 

-250 

-237 

-459 


Type IV. The determination of contours along 
which the radio gain (or A) is constant (the coverage 
problem): A radar has a wavelength of 0.107 meter 
and a power output of 750 kw. Assume a receiver in 
space with a minimum detectable power of 1CT 10 
w’att. The maximum possible distance between the 
radar transmitter w r hose elevation is 100 meters and 
the receiver for varying heights of the receiver is to 
be found. The gain of the radar antenna is 10,000 
(or 40 decibels), the gain of the receiver will be 
assumed to be 30 decibels. 

For the radar problem, a target of radar cross 
section a = 50 square meters is assumed to take the 
place of the receiver. The minimum detectable 
power of the radar is taken as 10" 10 v r att; the range 
of the set for varying altitudes of the target wdll be 
calculated. 

One-way: The radio gain sought is the ratio 
of the minimum detectable power to the powder 
output, or 


P 2 _ 10~ 10 

Pi 750 X 10 3 


4 

3 


X 10 1 - 16 , 


or 

10 log (P 2 /Pi) = -160+ 1 = —159 db. 


From equation (3), 

20 log A = - 159 - 40 - 30, 
= - 229 db. 


From Figure 41, 

20 log gi = 18.5, 
20 log hi = 40. 


Therefore 

20 log I = -229 - 40 - 18.5, 

= -287.5 db. 

Referring to Figure 37, the pairs of values of sd and 
eh 2 along the contour 20 log A = —287 are given 
in Table 7. By means of Figures 40 and 41, sd and 
eh 2 are changed to d and h 2 . The points found are 
to the right of the curve for ehi = 7.3, so that they 
correspond to points in the diffraction region. 


Table 7 


d/cm 

sd 

ehi 

h -2 meters 

118 

11 

1.3 

18 

129 

12 

3.5 

48 

140 

13 

7.0 

96 

151 

14 

11.5 

158 

161 

15 

16.5 

226 


Radar: The value of 10 log (P 2 /Pi) is the same as 
for the one-wvay calculation, —159 db. This must 
be changed to 20 log A by equation (5), 

20 log A = —141 db, 
and, as above, 

20 log I = -141 - 40 - 18.5, 

= -199.5 db. 

Referring to Figure 37, w r e see that the contour 
20 log A = —199.5 is to the left of ehi = 7.3 [see 
caution in equation (181)]. Therefore it is not possible 
to get the necessary power return P 2 from the given 
target so long as it is below the line of sight. The 
desired contour w T ould lie above the line of sight. 
The determination of the contour is discussed in 
Section 5.6.5. 

5 7 4 Sea Water, VHF, 

Vertical Polarization 

1 . Graphical Aids. Graphical aids are given in 
this section, w T hich, as in Section 5.7.3, are valid for 
all practical distances w'hen both antennas are low 
[h < h c (see Figure 35) ] and 2 hji 2 < < \d. If one 
or both antennas are elevated, they will give the 
value of the radio gain for the first mode, w'hich is a 
good approximation for the result found by summing 
all the modes w 7 hen the receiver is well within the 
diffraction region, i.e., w hen d > d L . [If one antenna 
is low ( h < h c ) and one well elevated (h > 40 h c ), 
the result found by using several modes is given in 
Section 5.7.5.] 




















BELOW THE INTERFERENCE REGION 


109 


Referring to equation (162) and Section 5.7.1, 

A = 2A 0 A 1 F 8 (H L gg') 1 (H L gg') 2 , (184) 

with gg' = 1 for h < h c . 
h c is given by Figure 35, 

20 log gg' is given by Figure 36, 

20 log A 0 is given by Figure 3 in Chapter 2, 

F a is given by Figures 31,,32, and 33. 

2. Plane earth factor A i. This has been discussed in 
Section 5.7.1. Ai is a function of p'd) p' is given in 
Figure 44 and 20 log A\ in Figure 45. The curve 
shift of Ai with X in the VHF band is less than 1 db. 
When p'd > 50, A\ = l/p'd, as in the dielectric 
ease. 

3. The low height-gain H L is a function of Ih (see 
Section 5.7.1). I is given by Figure 46, H L or 20 log H L 
by Figure 47. H L depends on the curve parameter 

Q = € r /60<rX wdiich for sea tvater is —. Since 

3\ 

H L —>lh for Ih > 4, 20 log H L can be found from 
Figure 34. 

4. h h h 2 > 4/1 and d > 50 /p'. As in Section 5.7.1 
[noting especially equations (151) and (153)], 
equation (184) reduces, when h > 4/1 and d > 50 /p', 
to 

A (185) 

2 Gr 


Table 8. Values of 4/1 and 50/p' for various wavelengths. 


X 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

m 

4 

l 

3 

27 

50 

80 

110 

143 

174 

222 

267 

308 

m 

50 

V ' 

2 

17 

17 

30 

50 

71 

100 

125 

175 

200 

km 


Equation (185) may be used generally, provided 
(gg'h) is replaced by H L /l when h < 4/1, and the 
right-hand member is multiplied by Aip'd or, in 
decibels, 20 log Ai + 20 log p'd are added, when 
d < 50 /p'. In this formula d < 50 /p' is given in 
meters. 

5. Horizontal versus vertical polarization. It is of 
interest to compare the gain of vertically and hori¬ 
zontally polarized waves over sea water at VHF. 
(It has been pointed out earlier that there is no 
marked difference in attenuation between horizon¬ 
tally and vertically polarized waves for wavelengths 
less than one meter.) Equation (184) is valid for 


horizontally polarized waves also by using the 
appropriate F 8 curve and putting g' = 1 and can be 
made the basis of a comparison between vertical 
and horizontal polarization. See, for comparison, 
equation (160). 



Figure 44. p' versus X for sea water, vertical polariza¬ 
tion. 



Figure 45. Plane earth factor A i, for sea water, 
vertical polarization. Note: All db values are negative. 




































































































































































110 


CALCULATION OF RADIO GAIN 


FREQUENCY IN MEGACYCLES 

4000 2000 400 200 40 



+ 20 


+ 10 


0 


-10 


-20 


-30 


-40 


-50 


-60 


Figure 46. Parameter l versus frequency for vertical polarization. 


20 LOG / 

















































































































BELOW THE INTERFERENCE REGION 


111 


For antennas at, or very close to, zero height, the 
gain-factor ratio depends on AiF 8 . While F s gives 
greater attenuation (lower gain) for horizontal than 
for vertical polarization, the difference between the 
two lies principally in the values of Ai. For X = 1 
meter, the ratio is 64,000 to 1 in favor of vertical 
polarization. For X = 10 meters, the ratio is about 
8.6 X 10 6 . 

However, as the antennas are raised above the 
ground, the strength of a horizontally polarized 
field increases much more rapidly than does the 
corresponding vertically polarized field, for a given 


which, as in Section 5.7.3, can be written 

A = 1.77 X lO-ffAWi, (188) 

L(sd) 2 J 

or 

20 log A -135 + 20 log + 20 log A 

(. sd ) 2 

+ 20 log p'd + 20 log i p. (189) 

Since F s has a graphical representation which de¬ 
pends on the wavelength, it is necessary to assume 
a particular value of X; equation (188) then becomes 



!h 

Figure 47. Height-gain function H l versus Ih, for low antenna heights. [See equation (152.)] 


wavelength up to a certain height above which the 
field is substantially independent of polarization. 
For example, above a height of 3 meters for X = 1 
meter, and above 77 meters for X = 10 meters, the 
two fields are practically equal. 

6. Parameter A. As in Section 5.7.3, curves can 
be drawn in terms of the parameter A where 


A = 


A 

(hgg')i 


for hi > 4/1, 


Al 
. H l 


for hi < 4/1. 


(186) 


Equation (185), including the correction for 
d < 50/p', becomes 

A = Wh\Wh\, (187) 

Z ur 


a relation between A, di, and h 2 , and Figures 48, 49, 
and 50 for X = 1, 3, 6 meters are in terms of these 
coordinates. The height-gain function of the trans¬ 
mitter gi can be found from Figure 41. 

7. Illustrative example: Communication. A com¬ 
munication set used in ship-to-ship work has a wave¬ 
length of 1 meter, a receiver sensitivity of 10 micro¬ 
volts with a resistance of 50 ohms across the input 
terminals and a transmitter power output of 100 
watts. The transmitter and receiver antennas are 
vertical half-wave dipoles at an elevation of 30 meters. 
The range is to be found. 

To produce a voltage of 10 microvolts across 
50 ohms, a power of 

p = Y1 = 100 x 10 ~ 12 

R 50 


watt = 2 X 10 12 watt, 
































































5000 

2000 

1000 

500 

200 

100 

50 

20 

10 

5 

2 

I 


CALCULATION OF RADIO GAIN 



d IN KILOMETERS 


Figure 48. Maximum range for X = 1 meter, vertical polarization. [See equation (186).] 








































































METERS 


BELOW THE INTERFERENCE REGION 


113 



d IN KILOMETERS 

Figure 49. Maximum range forX = 3 meters (sea water vertical polarization — 20 log g' = 1). See equation (186). 
























































IN METERS 


114 


CALCULATION OF RADIO GAIN 



Figure 50. Maximum range for X = 6 meters, vertical polarization. See equation (186). 




































































BELOW THE INTERFERENCE REGION 


115 


so that this is the minimum detectable power. The 
value of P 2 /P 1 for the given power output of 100 
watts is 


2 X 10 - 12 
100 


= 2 X 10 - 14 , 


and 


10 log— = 3 + (-140)-137. 

Pi 


This is to be changed to 20 log A by equation (3). The 
gain of a half-wave dipole over a doublet is 1.09 
(see Sections 2.2.2 and 3.2.3), so that Gi = G 2 = 1.09 
or 0.4 db, 

20 log A = -137 —0.8 = — 138 db. 

In changing from 20 log A to 20 log A it must be 
determined which of the relations in equation (186) 
is required by comparing the transmitter height of 
30 meters with 4/1. The value of Z as given by 
Figure 46 is 0.4. Hence the value of 4/Z is 10, which 
is less than 30. Then 

20 log A = 20 log A - 20 log 30 - 20 log gg', 

= - 138 - 30 - 0, 

= - 168. 

Referring to the chart for X = 1, Figure 48, we 
iind that for h 2 = 30 meters and 20 log A = — 168, 
the distance d is 53 kilometers. This then is the 
maximum theoretical range between the two sets. 


5,7,5 Radio Gain Near the Line of Sight 

For d much greater than d L , the first mode is 
sufficient, as given in Sections 5.7.3 and 5.7.4. For 
d nearly equal to d L , i.e., the receiver near the line of 
sight, a formula [equation (190)] can be given which 
takes into account several modes and still permits 
the use of graphical aids. This formula is valid only 
when the elevated antenna is very high, i.e., 
h > 1,200\ 2/3 and the other antenna is low, i.e., 
h < 30X 2 3 . (Otherwise the transition curve near 
the line of sight must be sketched in graphically, as 
indicated by the broken portion in Figure 7.) 

Denoting by H L 1 the height-gain of the low antenna 
at height hi, 

A = 2A 0 H Ll M{&) ("^-TV 2 (A) (190) 

L 2eh 2 J 

where h 2 and F 2 { A) refer to the elevated antenna. 


1. sd and eh are given in Figures 40 and 41. 8 is 
given in Section 5.7.6. 

2. A = V^(5) (sd - V2 eh,). 

3. g(8) = 1, except for the VHF range, vertical 
polarization, over sea water. The values are given in 
Table 9. 


Table 9. Values of V g(f>) for VHF (sea water). 


X 

I 1 234567 89 10 meters 

\V(<5) 

0.98 0.97 0.95 0.94 0.93 0.91 0.90 0.88 0.86 0.84 


A = ifglJj (sd-|/2eh 2 ) 



Figure 51. F(A) [representing /'’ 2 (A)] versus A for use in 
equation (190). 


4. F 2 (A) is given by Figure 51. 

5. M(8) for vertical polarization is given by 
Figure 52 as a function of 8 which can be found from 
Figures 53 and 54. 

For horizontal polarization 


M(8) = 




(Jl Y _ 1 

\2vkuJ I )- + (60(rX) 2 


• ( 191 ) 















































































116 


CALCULATION OF RADIO GAIN 


5 7 6 General Solution for Vertical 
(or Horizontal) Dipole 
Over a Smooth Sphere 

1. Field strength of dipole. The vertical component 
of the electrical field of a vertical dipole radiating 
in a homogeneous atmosphere over a sphere of 
radius ka (or horizontal component in the case of a 
horizontal dipole) is given by equation (192). The 
solution is valid provided the distance between 
receiver and transmitter and the radius of the 


For horizontal polarization, 

a \ 2/3 


-(*?)’ 


(«c - 1) 


14.2X10* 7 4 

^ 273 — ( € c - 1) for k = -. 


d. f = sd 
where 


= ( 2 *• V 73 

\\k 2 a 2 / 



sphere are much greater than a wavelength, condi¬ 
tions which are fulfilled in any practical application 
of short waves. 


E = 2E 0 (2 t r f) 


1/2 


s + 2t * 


■ (192) 


a. hi and /i 2 are antenna heights, 

b. E 0 is the value of E for a doublet in free space, 

c. 5 is the ground parameter which depends on 
the complex dielectric constant e c = e r —j60<j\. 


8 


For vertical polarization, 
= (2*kaj /s e c - 1 


14.2 X 10* € c - 1 

X 2/3 


for k = -. 
3 


(193) 


or 

s = 4.43 X 1(TV I/3 (—y 3 
\3 k) 


(194) 


e. t u are complex numbers which characterize 
the individual terms (modes) in equation 
(192). They are a function of 8. 

f. f n (hi) and f n {hf) are height-gain functions 
for the nth mode. 


2. The complex ground parameter 8. 8 depends on 
the wavelength and the electrical constants of the 
ground. (The dielectric constant is referred to air 
as unity.) 8 is large for horizontally polarized waves, 
but for vertically polarized waves it may vary con¬ 
siderably, as may be seen from Figure 53. In Figure 
54, the phase of 8 is given. For wavelengths less 











































































































BELOW THE INTERFERENCE REGION 


117 



Figure 53. | 8 | versus X for vertical polarization (see Table 10). Dielectric earth, 8—> oo ; perfect conductor, 8—> 0. 

Phase of 8 is 45° for intersection of two asymptotes for any particular ground. 











































































































































































118 


CALCULATION OF RADIO GAIN 


than 10 meters | 5 |, as given by Figure 53 for vertical 
polarization, is large except in the VHF range over 
sea water, e r = 80, <r = 4. 

The ground constants e r and a for water and 
various types of earth are given in Table 10. For 
X < X c the ground material is a dielectric earth. 


Table 10. Ground constants. 


Type of ground 

X — * r 

C 60 or 

Sea water 

s 

II 

<r = 4 mhos per meter 

0.33 m 

Fresh water 

er = 80 

<r =5 X 10~ 3 

267 

Moist soil 

e r = 30 

<r = 0.02 

25 

Fertile ground 

e r = 15 

<r = 5 X 10" 3 

50 

Rocky ground 

= 7 

<r = 10- 3 

117 

Dry soil 

er = 4 

<r = 10- 2 

6.7 

Very dry soil 

er =4 

<r = IQ" 3 

66.7 


3. The mode numbers r n . These numbers are given 
below for the two limiting values for 5 = 0 (infinite 
conductivity or X —» oo ), and for 5 = oo , i.e., the dielec¬ 
tric earth. 


Mode 

No. 

5=0 

5 = oo 

1 

r 1>0 = 0.885 e ' Jir/3 

Ti ><a = 1.856 e~ jn/3 

2 

t 2 .o = 2.577 e~ jn/s 

T 2 ,flo = 3.245 e' jw/s 

3 

r 3 ,o = 3.824 e~ jn ' 3 

t 3i oo = 4.382 e~ jir/s 

>4 

If n>4, 

t„.o = i[37r(n+l)] 2 /V J,r/3 

If n > 4, 

T «,«o =i[37r(n+I)]2/V^/ 3 



Figure 54. Phase of 5 for vertical polarization. See Table 10. 





















































































BELOW THE INTERFERENCE REGION 


119 


From these limiting values of r the value of r in 
general for any given 8 can be found from the follow¬ 
ing two series, of which only the first is of interest 
in short-wave work. 


8 large, 

= ^n.ao * 

+ • • * • 





(195) 


8 small, 


8 1/2 8 



4. The height-gain Junctions J(h). 

(a) For low antennas (h < 30\ 2/3 ), the height-gain 
functions, to the first approximation, are inde¬ 
pendent of n. 


m = i + j 

m = i +j 




for vertical polarization, 
(196) 

for horizontal polarization 


Note that the magnitudes of the bracketed quantities 
are equal to Ih. The magnitude of / has been denoted 
by H l and is represented in Figure 47 as a function 
of Ih. The phase of the bracketed quantities in 
equation (196) is taken into account by using differ¬ 
ent curves with the parameter Q = e r /60crX. For 
large values of lh,H L — >lh. 

(b) For elevated antennas (h > > 30\ 2/3 ). The 
function f„ can be represented by 


3 1 exp { + jj - j\ [(eh) - 2 r„] 3/2 } 

V2?r (2 eh) 1 '* Ji/z(x) + J- 1 / 3 O) 

(197) 


where, from equation (159), 

_ (4 ^) 1/3 _X - 2/3 /£\ 1/3 
6 (\ 2 ka) 1/3 60 \3k) 


(198) 


and where the argument of the two Bessel functions 
is 


x = i(~ 2 r „) 3/2 e" 1 ” 

For the nth mode, if (eh) >> 2r„, the magnitude of 
f„ can be written 


3 1 exp(jT„^2eti) 

V2^(2 eh) l/i J 1/3 (x) + J- U3 {x)' 


For large 8, using the first two terms of equation 
(195), and writing x n for x when r» is replaced by 
r », 00 > 

/— 2 t y /2 

x = x -(rrV ■ 

Substituting this in /i / 3 (x) + /_ i/ 3 (x), writing down 
the first two terms of the Taylor expansion, making 
use of the fact that the r » t00 are roots of Ji/$(x) + 
J-i/z(x) = 0 and of the relation given by a prop¬ 
erty of the Bessel function, 


J\/s( x ) + J-\/s( x ) — ~ 1 /(3a?) \J\ /3 (%) + J-i/3( x )] 

J-2/z( x ) ~ J2/z( x )> 

we have 

J\/z( x ) "b J- 1/30*0 

/ \ 1/2 

— ~ \J-2/s( X ' D ) — ^2/300] • 

( 200 ) 


If these results are substituted into equation (192) 
for both antennas, the factor (5 -f 2r„) becomes 
1 + 2r n>ao /5, which approaches unity for large 8. 
This means that if both antennas are sufficiently 
elevated, short-wave propagation is practically inde¬ 
pendent of ground constants. 

The value of / given by equation (199) can be 
w r ritten as glh so that g represents the gain over Ih, 
the value approached by H L , when Ih > 4. The value 
of g for 8 —»oo is represented in Figure 36. 

If 8 is not very great, as in the case of vertically 
polarized VHF over sea water, the effect of 8 can be 
taken into acocunt by changing e to eg(8) and g to 
gg r . The functions g(8) and g f are given by Figure 55. 

5. Plane earth gain factor and shadow factor. The 
field near the ground over a plane earth with infinite 
conductivity is equal to 2Eo, twice the free-space 
field. For an imperfectly conducting ground, the 
field for antennas at zero height over a plane earth 
may be written , 

E = 2EqAi. (201) 

Ai is represented in Figure 56 as a function of p'd, 
w r here 

_ 2x |U - 11 _ 2 t V(< r — l) 2 + (60(A) 2 

X 1 ( c 1 2 X € r 2 + (60crX) 2 


for vertical polarization. For horizontal polarization. 


2 7T 


2tt 


p' = y I € C -1 I = — V(e r — 1) 2 + (60<rX) 2 . (203) 


The curve parameter is Q = « r /60<rX. 


( 199 ) 















120 


CALCULATION OF RADIO GAIN 


Comparing equations (202) and (203) with 
equations (193) and (194), we find that 

p'd = | S | r. (204) 


height [i.e., /(0) = 1] and equation (205) is now of 
the form 

E = 2E 0 A 1 F a (208) 


Hence, equation (192) may be written as 


5 - 3/2 

E = 2^o (2 7r) 1/2 -— 
p'd 


s 


e- iT n s 
1 + 2 tJS 


fn(hl)f n (hd 


(205) 


If & is large (e.g., X small), 2r„/<j in equation (207) 
may be neglected and r„ replaced by ^ so that the 
shadow factor is practically independent of ground 
constants. The shadow factor is represented graphi- 



Figure 55. 


V <7(5) and g' as functions of magnitude and phase. 


If p'd is large, we see from Figure 56 that 



(206) 


so that the physical significance of 1 /p'd in equation 
(205) becomes apparent. 

The factor 


F 8 


(27r) 1/2 r 3/2 


e jT n* 

1 + 2r n /8 


(207) 


represents the effect of earth curvature in increasing 
the attenuation over that of a plane earth at zero 


cally in Figures 32 and 58. Where the factor 2r n /5 
cannot be neglected, as in the VHF range, vertical 
polarization over sea water, the dependence of 
F s on 5 is accommodated by changing the abscissa 
from £ = sd to V = s'd where s' = s/(5), and by 
representing F s by a family of curves in Figure 58, 
whose parameter is given by dotted lines in Figure 
57. /(5) is represented also in Figure 57. For V < 0.4, 
F a is less than 1 db below unity. This corresponds 
to a distance over which the earth may be considered 
plane, i.e., d < 10 4 X 1/3 , as given in Section 5.7.1. 
The greater the wavelength, the smaller the effect 
of the earth’s curvature. 





































































































































































BELOW THE INTERFERENCE REGION 


121 


o in 2 £ 8 

I » " r ~ » 1 I 1 " ■ I 1 11 ' I i ■ i i I ' 


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p , d = (2'TTd/X) , \J(€ r -|) 2 +(60 a>.) 2 FOR HORIZONTAL POLARIZATION 
Figure 56. Plane earth gain factor A x versus p'd. A, = l/(p'd) for p'd >50. 


























































































































































































































































122 


CALCULATION OF RADIO GAIN 


Perfect Conducto* 


Dielectric Eart h 



Figure 57. f(8) versus 8. Solid lines correspond to phase of 8. Dotted lines indicate curve in Figure 58. For horizontal 
polarization f(8) = 1. 



Figure 58. Shadow factor F s versus 77 = f/(5) = sdf(8). See Figures 32, 33, and 57. Curve +10 corresponds to a 
perfect conductor. 












































































































































































BELOW THE INTERFERENCE REGION 


123 


function of distance d for doublets at zero height. 
Figure 60 gives the first mode height-gain factors for 
transmitter and receiver heights, hi and h 2 , respec¬ 
tively. 

To obtain the radio gain under different conditions 
it is merely necessary to add the decibel gains of the 
transmitter and receiver antennas, the radio gain for 
zero height (Figure 59), the height-gain factor 
(Figure 60) for the transmitter, and a similar figure 



100 1,000 10,000 100,000 1 , 000,000 10 , 000,000 

DISTANCE d IN METERS 

Figure 59. Free-space radio gain A 0 (-) and radio gain A, in decibels, for propagation over very dry soil with 

doublet antennas on the ground (-for horizontal polarization and-for vertical polarization). Numbers on the 

curves give the wavelength X. Note: Radio gain is independent of the radiated power. 


If the antennas are elevated, F s can still be used 
for the first mode for great distances where the first- 
mode gives most of the field. 

5.7.7 Sample Calculation 

for Very Dry Soil 

The general solution given in Section 5.7.6 is here 
illustrated for the case of doublet antennas, either 


horizontally or vertically polarized, placed at various 
heights over an earth assumed to be very dry soil 
for which the constants are e r = 4, and a = 0.001 
mho/meter. The following graphs cover, in decimal 
steps, the frequency range of 30,000 to 0.03 me or 
wavelengths X = 0.01 to 10,000 meters. 

Figure 59 gives the free space radio gain A 0 and 
the radio gain A decibels over very dry soil, as a 


for the receiver (Figure 60). This process, however, is 
subject to the restriction mentioned in the next- 
paragraphs. 

The addition of the factors given in the preceding 
paragraph is valid all the way up to the maximum of 
the first lobe, where the field is given by the sum of the 
direct and reflected rays, provided that the antennas 
have comparable heights. 






















































































































124 


CALCULATION OF RADIO GAIN 


The radio gain can therefore never be more than 
6 db greater than the free-space gain with the same 
antennas. 

If, however, one antenna is low, h < h c , and the 
other is very high, h > 40 h c , the method discussed 


four tables of computations are given. Table 11 
gives the values of certain quantities, for a wide 
range of frequencies and for very dry soil, which are 
independent of polarization. Those quantities which 
are dependent on polarization are given in Table 12. 



0.01 0.1 1.0 10 100 1,000 10,000 1,000,000 

HEIGHTS h, OR h 2 IN METERS 

Figure 60. First mode height-gain factors for transmitter and receiver. 


fails since the height-gain factors are based on the 
first mode only. In this event, either the methods 
outlined in Section 5.7.5 must be employed or the 
radio gain at low elevations must be connected 
graphically with the value obtained in the optical 
region for the first maximum. 

As an aid to the computer in checking his results, 


Table 13 gives detailed calculations for ground level 
radio gain for doublets for a wavelength of X = 1.0 
meter, while Table 14 gives the first mode height- 
gain factors for the same wavelength. 

Similar charts and tables may be prepared easily 
for transmission over other types of earth for a 
similar range of frequencies. 




















































































































































Table 11. Quantities independent of polarization. 

Earth, Very Dry Soil. e r = 4; a = 0.001 mhos /meter; e c = 4 — j0.06\ = 4(1 — j0.015\); e c — 1 = 3 — j0.06\ = 3(1 — j0.02X). 


BELOW THE INTERFERENCE REGION 


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128 


CALCULATION OF RADIO GAIN 


Table 14. Height-gain factors (first mode). 

Earth, Very Dry Soil. X = 1.0 m; b H = 0.426 X 10 6 (- 1.1°); 8 V = 2.6 X 10 4 (0°); l R = 10.86; l v = 2.72. 

First mode height-gain factors, h c = 30. m; e = —— = 0.01667; Q = 66.67. 

2h c 


h 



Hl 


g(8) 


g 

g' 




Polar- 

Ih 


eh 


ehg (5) 

Eq. (157) 


H L gg' 

20 log (H L gg') 

m 

ization 


Fig. 47 


Fig. 55 


Fig. 36 

Fig. 55 



0.01 

H 

.1086 

1. 




1 

1 

i 

0 

V 

.0272 

1. 




1 

1 

l 

0 

0.1 

H 

1.086 

1.5 




1 

1 

1.5 

3.5 

V 

.272 

1. 




1 

1 

1 

0 

0.25 

H 

2.725 

3. 




1 

1 

2.73 

8.68 

V 

.68 

1.25 




1 

1 

1.25 

1.9 

0.5 

H 

5.43 

5.5 




1 

1 

5.5 

14.8 

V 

1.36 

1.7 




1 

1 

1.7 

4.6 

1. 

H 

10.86 

10.86 




1 

1 

10.86 

20.7 

V 

2.72 

2.8 




1 

1 

2.8 

8.9 

Q 

H 

32.58 

32.58 




1 

1 

32.58 

30.3 

O 

V 

8.16 

8.16 




1 

1 

8.16 

18.2 

10. 

H 

108.6 

108.6 




1 

1 

108.6 

40.7 

V 

27.2 

27.2 




1 

1 

27.2 

28.7 

30. 

H 

325.8 

325.8 

0.5 

1 

0.5 

1 

1 

325.8 

50.3 

V 

81.6 

81.6 

1 

0.5 

1 

1 

81.6 

38.3 

100. 

H 

1,086. 

1,086. 

1.667 

1 

1.667 

1.5 

1 

1,630 

64.2 

V 

272. 

272. 

1 

1.667 

1.5 

1 

408 

52.2 

300. 

H 

3,258. 

3,258. 

5. 

1 

5. 

4.3 

1 

14,000 

82.9 

V 

816. 

816. 

1 

5. 

4.3 

1 

3,510 

70.9 

1,000. 

H 

10,860. 

10,860. 

16.67 

1 

16.67 

100. 

1 

1.086 X 10 6 

120.7 

V 

2,720. 

2,720. 

1 

16.67 

100. 

1 

0.272 X 10 6 

108.7 

2,000. 

H 

21,720. 

21,720. 

33.33 

1 

33.33 

1,585. 

1 

3.44 X 10 7 

151. 

V 

5,440. 

5,440. 

1 

33.33 

1,585. 

1 

0.863 X 10 7 

139. 

5,000. 

H 

54,300. 

54,300. 

83.33 

1 

83.33 

1.497 X 10 6 

1 

8.13 X 10 10 

218. 

V 

13,600. 

13,600. 

1 

83.33 

1.497 X 10 6 

1 

2.035 X 10 10 

206. 
































Chapter 6 

COVERAGE DIAGRAMS 


61 DEFINITIONS 

T he locus of points in space having a constant 
field strength is called a coverage diagram. In 
the optical region this is also called a lobe diagram. 
The construction of these diagrams is an important 
part of the predetermination of the performance of 
radar and communication sets. The basic concepts 
and formulas will be developed first for the case of 
the plane earth and then applied with necessary 
modifications to propagation over a spherical earth. 

The method outlined in this chapter is applicable 
only to the lobe structure lying above the tip of the 
first lobe. In this region the field is given almost 
entirely by the vector sum of the direct and reflected 
waves. The lower portion of the first lobe is dis¬ 
torted from the regular lobe structure because, in 
this region, the field strength is determined in part 
by contributions from the diffraction terms as well 
as by the contributions of the direct and reflected 
waves. 

62 PLANE EARTH 

6,2,1 Field Strength 

For horizontal polarization and a reflection coeffi¬ 
cient equal to —1 (i.e., p = 1, <j> = 180°), the re¬ 
ceived field intensity oscillates from zero to twice 
the free-space value, depending on the position of 
the point in space, as shown in Figure 1. The posi¬ 
tion in space determines the path difference A, 



Figure 1. Coverage diagram for plane earth (heights 
h 2 are exaggerated relative to distance d ). n = 1,3,5 
.... for the first, second, third .... lobes, d = dmax 
sin (xn/2) and d ma x = 2 d 0 . 

which in turn determines the phase retardation, 
5 lag , due to path difference. The angle 12/2 used in 


calculating E by equation (29) in Chapter 5 is a 
function of 5 lag and </>, since 12 = 5 lag +</>'. The 
effect of a reflection coefficient p less than unity is to 
reduce the length of the lobe maxima to values less 
than 2d 0 and to increase the minima above zero as 
indicated by the dotted lines of Figure 1. The angles 
at which the maxima and minima occur depend 
upon the phase shift at reflection, as will be explained 
in the following section. 

6 2 2 Angles of Lobe Maxima 
(Horizontal Polarization) 

Lobe maxima occur whenever the sum of the phase 
shifts caused by reflection and path difference equals 
an even multiple of r radians, while lobe minima 
(nulls) occur when the total phase shift is an odd 
multiple of x radians. If p = 1, the nulls are equal 
to zero. 

It follows that for horizontal polarization 
(4> = 7 r), maxima occur when 8 equals x, 3 x, 
5x, etc., and minima when 8 =0, 2x, 47r, etc. 
This means that a path difference equal to an odd 
multiple of X/2 gives a lobe maximum while a path 
difference equal to an even multiple of X/2 gives a 
null. This holds only for horizontal polarization 
(</> = x). Applying equation (52) in Chapter 5 
to the case when di << d 2 (i.e., d 2 = d), 

A = ———— = 2/ij tan ^ ^ 2 h tan 7 = 27iiy, (1) 

d 

where y is the angle in radians between the hori¬ 
zontal and the line joining the receiver or target 
to the base of the transmitter (see Figure 8 in Chap¬ 
ter 5). For equation (1) to hold, \p must be less than 
0.2. Hence for maxima, 

0 , n\ 

2h iy =~, 

and for minima, 

cyr ™X 

2hy = -, 

or 


n = odd integer, 


( 2 ) 


n = even integer, 


129 






130 


COVERAGE DIAGRAMS 


and n has a range of 0 to 2 for the first lobe, 2 to 4 
for the second lobe, etc. 

The limitations of equation ( 2 ) may be summarized 
as follows: 

1. The phase shift at reflection is x radians. 
This assumes horizontal polarization, or \J/ = 0 for 
vertical polarization. 

2 . The reflection point is (relatively) close to the 
transmitter. 

3. The grazing angles corresponding to lobe 
maxima are less than 0.2 radians. In connection with 
limitation ( 2 ), it should be noted that the angles for 
which the approximation y = \J/ holds depend upon 
the wavelength and transmitter height. The follow¬ 
ing table shows the minimum angle for various 
transmitter heights and wavelengths at which the 
error in the path difference A introduced by this 
assumption is less than 1 per cent. To satisfy equa¬ 
tion ( 2 ), and have an error less than 1 per cent, 


Table 1 


Antenna 
height, hi 
(meters) 

y 

Minimum 

(degrees) 

nX 

Minimum 

(meters) 

Minimum n at 
X = 3 meters 

120 

1.44 

12.9 

4.3 

60 

1.1 

4.5 

1.5 

30 

0.78 

1.6 

0.53 

15 

0.56 

0.6 

0.2 

9 

0.4 

0.25 

0.08 


7 should lie between the minimum value and 0.2 
radians. Table 1 shows, for increasing transmitter 
antenna height, how the angle for which equation ( 2 ) 
is valid within 1 per cent also increases. 

If n is set equal to 2m — 1 , integral values of m 
correspond to lobe maxima and half-odd integers to 
lobe minima. The advantage of this notation is that 
the value of m is the number of the maxima or lobe 
number. Thus for the fifth lobe, m = 5. The gen¬ 
eral expression for the grazing angles corresponding 
to lobe maxima, for a plane earth and for horizontal 
polarization, is 


_ ( 2 m — 1 ) 
4 h 


where integral values of m give maxima and half¬ 
odd integers give minima. 


6 2 3 Angles of Lobe Maxima 
(Vertical Polarization) 

With vertically polarized radiation the reflection 
phase shift </> [equation (27) in Chapter 5] is less 


than 7 r radians (i. e., </>'=</> — x is negative). 
It follows that the path difference for lobe maxima 
must be greater than X/2 and greater than X for the 
nulls. In other words, the value of n in equation (2) 
must be increased by (A + n) to compensate for the 
decreased phase shift of reflection, so that 

(An) (y)(y) -(»-«)- 0. (4) 

Hence 

(An) = t - 1 = + i'. ( 5 ) 

X X 

For vertical polarization, equation ( 2 ) becomes 

y — . n - ± (An) . ( 6 ) 


Lobe Equation 


When p = 1, and </> = x, </>' = 0, and 12 = $, 
equation (46) in Chapter 5 may be written as 


12 8 

d = 2>IGido sin ~ = 2y/Gid 0 sin~. (7) 

Substituting 

g _ 

gives 

d = 2 V<?i d„ sin (-^p) > ( g ) 


where d 0 is the free-space range which may be com¬ 
puted from the gain corresponding to the given 
coverage diagram by use of the nomogram given in 
Figure 3 in Chapter 2 . Equation ( 8 ) may be written 
as 


d = 2y/Gid 0 sin 



sin 7 


) 


(9) 


Equation (9) shows that for fixed values of free- 
space range do, transmitter height hi, and wave¬ 
length, the coverage lobe may be represented by a 
polar sine function of the angle y at the base of the 
antenna. This assumes that the slant range measured 
to any point on the lobe may be considered equal 
to the distance d measured along the surface of the 
earth. 











SPHERICAL EARTH 


131 


If n in equation (2) is allowed to assume all frac¬ 
tional and integral values from 0 to 2, sin y —» y 
maj r be expressed as 

n\ /m\ 

sin 7 = 7 = — . (10) 

4hi 

Substituting this value into equation (9) gives 

d = 2VG ? id 0 sin = 2^Gid 0 sm (90°n). (11) 

Equation (11) is useful in sketching the lobe contour. 
It holds only when the reflection coefficient equals 
— 1 (i.e., p = +1) and the angle y is small enough 
so that equation (10) is valid. 


small null areas. The shape of the contour is, there¬ 
fore, less important and it is sufficient to find the 
maximum and minimum ranges and then to sketch 
the lobe from the polar sine formula [equation (11)]. 
On the other hand, the shape of the contour is of 
great importance when there are few lobes and the 
null area is large. This is illustrated in Figure 3, 
where there are only two complete lobes in the 
region of interest . 

The second point to be noted is the varying effect 
of divergence on the lobe number and angle. It 
may be seen from equation (89) in Chapter 5 that 
the divergence factor approaches unity as tan \p 





Figure 2. Vertical coverage diagram. 


63 SPHERICAL EARTH 

6,3,1 Lobe Characteristics 

Figures 2 and 3 are typical vertical coverage 
diagrams for a smooth spherical earth. They illus¬ 
trate two important points. 

The first is the dependence of the number of lobes 
on the ratio of transmitter height to wavelength. 
Figures 2 and 3 show that for hi equal to 75.4 wave¬ 
lengths, the lobes are much more closely spaced 
than for a transmitter height of 32.3 wavelengths. 

When the number of lobes is large, there is little 
possibility for a target to escape detection in the 


increases (see also Figure 11). The divergence 
factor is low for small angles and then approaches 
unity rapidly. This accounts for the reduced range 
of the first three lobes of Figure 2. For the larger 
angles, the maximum range is approximately equal 
to twice the free-space range. When the ratio h\/\ is 
small, the angle at the first lobe maxima is large, since 
7 = n\/4:hi. In this case, the effects of divergence 
will be negligible except for the lower part of the 
first lobe, and the polar sine function derived for the 
plane earth may be used. 

There exists also the intermediate case where the 
effects of divergence may not be neglected and an 









132 


COVERAGE DIAGRAMS 


accurate knowledge of the lobe shape is required. 
Three different solutions of this problem are given 
in Sections 6.4, 6.5, 6.6, and 6.7. They are 

1. The p-q method for horizontal polarization only. 

2. The u-v method, which may be used for both 
horizontal and vertical polarization. 

3. Lobe-angle method which has the advantage of 
determining the lobe angles directly and is used for 
either polarization. 


where D replaces K in equation (46) in Chapter 5, 
since for horizontal polarization p = 1 and since 
we are neglecting any effect due to the antenna 
radiation pattern. For this expression, D and 12 are 
certain functions of the antenna heights hi and h% as 
well as d which were considered earlier (see Sec¬ 
tions 5.2.4 and 5.5.6). For a given transmitter, h h 
Gi, do, and X are given, so that for a given gain 
contour the only variables in equation (12) are d and 



0 32 64 96 128 160 192 224 256 288 320 

<J DISTANCE IN KILOMETERS 


Figure 3. Vertical coverage diagram. 


<>.4 THE p-q METHOD 

(HORIZONTAL POLARIZATION) 

6,41 Outline of Method 

This method consists in plotting the locus of 
points having a constant range d and locating those 
points on this curve which are at such a distance 
from the transmitter that the phase shift caused by 
path difference corresponds to the required range. 
The range corresponding to a total phase shift 12 
is given by 

d = Vcido yj (1 - D) 2 + 4 D sin 2 1, (12) 


/i 2 . The difficulty of the problem consists in the fact 
that equation (12) provides an extremely complicated 
relation between h% and d which cannot be solved 
explicitly for either coordinate. 

Under such circumstances, the natural procedure 
is to introduce new coordinates which make the 
handling of equation (12) easier. The method de¬ 
scribed in the following makes use of the variables 

d\ , d% 
p = — and q = — 

d T d 

discussed in Sections 5.5.5 and 5.5.7, and the pro¬ 
cedure will be called the p-q method. 







THE p-q METHOD 


133 


then proceed in the following manner. To a fair 
approximation, we may assume the extreme range 
of the lobes to correspond to sin 2 (12/2) = 1, so that 
by equation (12) the corresponding distance d max is 
given by 

4ax — ^Gi do(l + D). (16) 

Expressing d max and D by p and q, the above equation 
determines the envelope of all lobe maxima. The 
practical way of doing this is to use a graphical 
representation of D in p and q coordinates (Figure 17 
in Chapter 5) and to start by selecting a particular 


15 1 4 0=99 



Figure 4. Curves of constant-divergence factor D and path difference parameter R. (Radiation Laboratory.) 


It may be recalled that expressed in coordinates 
p and q 

d ~ V • d T , 

1 - q 

(13) 

z>=fi+ r /2 , 

L (i - p 2 )J 

(14) 

and the total phase shift, by equations 

129) in Chapter 5, is 

(97) and 

a _w q a-^ +4> , 

X dr p 

(15) 


For horizontal polarization, <j> —> 0, so that for this 
case (which is the one under consideration) all 
variable quantities in equation (12) have been 
expressed in terms of p and q. 

6 4 2 Construction of Range Loci 

Suppose to start with that we want to compute 
the position of the extreme range of a lobe. We may 


value of D, say D = D x . Inserting this in equation 
(16) gives a corresponding value of d max , and insert¬ 
ing this value of d max for d in equation (13) determines 
a straight line in the p,q plane, since dr = V 2kahi 
is known. Whatever is the value of d max , this line 
passes through the point q = 1, p = 0. In order 
to determine the position of the line, only one more 
point is needed. A convenient point to choose is to 
take q = 0.9 and compute the corresponding p from 































































134 


COVERAGE DIAGRAMS 


p = 0.1 d m& Jd T . The point of intersection of this line 
with the selected Di-contour then gives the desired 
p,q combination that corresponds to the given 
values of d max and Di. 

From this pair of values ( p,q) } the corresponding 
receiver height h 2 may be calculated by the relation 
[equation (98) in Chapter 5] 



6 4 3 Construction of 

Path-Difference Loci 

An assumed value of 12 in equation (12) determines 
5, the phase shift caused by the path difference, as 
5=12 + 27m, (18) 

where n assumes all integral values and zero. This 
value of 5 determines the path difference A = r — r d , 



Now both coordinates of the desired point are known, 
and the point may be plotted. Plotting a series of 
different points by the same method yields a smooth 
curve, which is the envelope of all lobe maxima. 

The locus of minima may similarly be plotted by 

using sin 2 — = 0 and 
2 

dmin = ^l0ido(l ~ D) (17) 

instead of equation (16). Intermediate range curves 
are found by assigning nonintegral values to m in the 
equation 12 = mir and substituting in equation (12). 


But from equation (97) in Chapter 5 


2fti 2 (1 - p 2 ) 2 
d T p 



( 20 ) 


where/(p) is given in Figure 18 in Chapter 5. 

In this calculation, q may be taken as the inde¬ 
pendent variable. The assumed values of q together 
with A from equation (19) determine /(p) in equa¬ 
tion (20). For given values of /(p), the correspond¬ 
ing values of p may be read from Figure 18 in Chap¬ 
ter 5. The coordinates h 2 and d on the path-differ- 













































THE u -V METHOD 


135 


(46) in Chapter 5] are constructed and their inter¬ 
sections with path-difference contours corresponding 
to the assumed values of 12/2 determine points on 
the lobes. 


6 5,2 Construction of Lobes 
(Horizontal Polarization) 

In this method, the divergence factor D is con¬ 
sidered to be the independent variable. Dividing 



Figure 6. Curves of constant-divergence factor D and path difference parameter R. (Radiation Laboratory.) 


ence loci are found from equation (98) in Chapter 5 
and equation (13) in Chapter 6, giving 



and 

i ~ dr (r^i) ■ 

Intersections of the path-difference loci with the 
range curves determine points on the lobes. 


6.5 THE u-v METHOD 

6.5.i Outline of Method 

This method makes use of the generalized coordi¬ 
nates u = h 2 /hi and v = d/d T described in Section 
5.5.8. The curves of constant-divergence factor D 
and path-difference parameter R are plotted on the 
same sheet in Figures 4, 5, and 6. The divergence 
lines are shown in full and the path-difference curves 
are dotted. Envelopes of constant sin 2 (12/2) [equation 


both sides of equation (46) in Chapter 5 by d T gives 

= V Gi do y (1 — K ) 2 + 4 K sin 2 — > (21) 

where 





























136 


COVERAGE DIAGRAMS 


If p = 1, and the effect of antenna directivity is 
neglected, K = D, and 

v = J- = \l(l - Dy + 4 D sin 2 —. (22) 

' 2 


given by Figure 12 in Chapter 5. Equation (22) 
then gives the value of v. These quantities together 
with R = nr may conveniently be tabulated, as in 
Table 2. Corresponding values of D and v are plotted 
as crosses on Figure 7. The line drawn through these 



Figure 7. D-v loci and lobes. 


The following discussion illustrates how one con¬ 
tour of a coverage diagram, corresponding to a 
particular value of radio gain, may be plotted on 
Figure 4 or its equivalent, Figure 7. The result is a 
curve similar to Figure 2, but plotted in u,v coordi¬ 
nates instead of h 2 and d. See also Figures 16 to 39. 

For illustration, let the transmitter gain, G\ = 1 and 
let the radio gain be such a value that do = d 0 /d T = 2. 
Further, let X = 0.1 meter and hi = 20 meters. 
From Figure 15 it is seen that r = l,030X//q 3/2 = 1.2. 
Select one of the curves for sin 2 (12/2) in Figure 12, 
Chapter 5, say sin 2 (12/2) = 1, for which 12 = it, Si r, 
5 ir, etc. These values correspond to tips of the lobes 
for which n = 1, 3, 5, etc., since, for perfect reflec¬ 
tion, 12 = nir by equation (116) in Chapter 5. 

Next select values of K = D and note the corre¬ 
sponding value of the radical 


yj( 1 - K)"- + 4A' sin 2 ~ 


Table 2. Values of v and R for sin 2 (fi/2) = 1, do =2. 


D 

yj(l - Dy+iDsin^ 
(Figure 12, Chapter 5) 

V 

[equation 

(22)] 

(r = 1.2) 

n 

R = nr 

Lobe 

maxima 

0.2 

0.3 

1.2 

1.3 

2.4 

2.6 

1 

1.2 

1st lobe 

0.4 

0.5 

1.4 

1.5 

2.8 

3.0 

3 

3.6 

2d lobe 

0.6 

0.7 

1.6 

1.7 

3.2 

3.4 

5 

6.0 

3d lobe 

0.8 

0.9 

1.8 

1.9 

3.6 

3.8 

7 

8.4 

4th lobe 

0.95 

1.0 

1.95 

2.0 

3.9 

4.0 





points is the locus of the tips of the lobes. The 
actual position of each lobe tip is then marked with a 
circle where the corresponding value of R crosses 
the locus in Figure 7. 




















































































THE u-v METHOD 


137 


Additional lobe points are located by choosing 
some other value of sin 2 (12/2), say sin 2 (0/2) = 0.7. 
Each value of sin 2 (0/2) now gives two points on each 
lobe, one on the upper branch and the other on the 
lower. Again choose values of K = D, obtain the 
corresponding values of the radical 

yj(l - K) 2 + 4K sin 2 - 

2 

from Figure 12 in Chapter 5, and calculate new 
values for v. The D-v values are plotted as crosses 
on Figure 7 and the line through them is the locus of 
points for which sin 2 (£2/2) = 0.7 (see Table 3). 


Table 3. Values of v and R for sin 2 {[1/2) = 0.7, do = 2. 


D 

yj{ 1 - D) 2 -h4Dsin 2 | 
(Figure 12, Chapter 5) 

V 


(» 

• = 1.2) 


K 

n 

R = nr 

Lobe 

0.2 

1.1 

2.2 

0 

0.63 

0.756 

1 

0.3 

1.15 

2.3 

0 

1.37 

1.644 

1 

0.4 

1.22 

2.44 

1 

2.63 

3.16 

2 

0.5 

1.28 

2.56 

1 

3.37 

4.04 

2 

0.6 

1.36 

2.72 

2 

4.63 

5.56 

3 

0.7 

1.43 

2.86 

2 

5.37 

6.44 

3 

0.8 

1.52 

3.04 

3 

6.63 

7.96 

4 

0.9 

1.6 

3.2 

3 

7.37 

8.84 

4 

0.95 

1.64 

3.28 





1.0 

1.68 

3.36 






For sin 2 (12/2) = 0.7, sin (£2/2) = ±0.836, £2/2 
= 0.3157T or 0.6857T. Then £2 = mr = 0.63x + 2A;x 
and 1.37 tt + 2/c7t, in which k is an integer. Then 
n = 0.63 + 2 k and 1.37 + 2k, and R = nr. Values 
of n and R are listed in Table 3. The intersections 
of the R values and the locus for sin 2 (£2/2) = 0.7 are 
plotted as circles on Figure 7. 

The entire lobe structure for one contour may be 
drawn by choosing additional values of sin 2 (£2/2). 
A large number of contours have been calculated 
by the Radiation Laboratory and are plotted in 
Figures 16 to 39. 

In order to construct one contour of a coverage 
diagram, it remains to find the intersection between 
the curves giving values of u for constant sin 2 (£2/2) 
and the corresponding path-difference contours. 
The equations relating R to £2/2 are given below. 
From equation (18) 


and from equation (19) 



From equation (83) in Chapter 5 
^ _ kaA 
hid ] 1 

An assigned value of £2 fixes two values of 8 for each 
lobe, as explained in the previous paragraph. All 
values of sin 2 (£2/2) other than 1 or 0 determine two 
intersections with the lobe. When sin 2 (£2/2) = 1, 
the envelope of maxima is obtained, while sin 2 (£2/2) 
= 0 corresponds to the envelope of minima. 

By selecting several values of sin 2 (£2/2) in Figure 
12, Chapter 5, and following the method outlined 
above, a coverage diagram may be constructed in 
generalized ( u,v) coordinates. The actual values of 
h 2 and d are 

h 2 = hiu, (24) 

d = d T v. (25) 


Construction of Lobes 
(Vertical Polarization) 


Problems involving vertical polarization or cases 
where the ratio of the antenna-pattern factors 
F 2 /F 1 cannot be neglected, may be solved by suc¬ 
cessive approximation. 

As a first approximation the method of Section 
6.5.2 is applied to determine points {h 2 ,d) on the lobe. 
The corresponding values of u and v determine s 
in Figure 19 or Figure 20, in Chapter 5, and tan0 
may be found from Figure 24 in Chapter 5 for the 
given transmitter height hi. An alternate method 
is to calculate tarn/' from equations (73), (58), and 
(60) in Chapter 5, which are 


di = sd, 



4 t K 
tani p = — . 

di 


The angles and v required to calculate the antenna 
pattern factors F x and F 2 are found from equations 
(62) and (63) in Chapter 5, 


tan^ d = 



d 
2 ka 




dx 

ka 


8 = £2 + 2tu (0 = 180°, 0' = 0), (23) 



























138 


COVERAGE DIAGRAMS 


The values of p, </> (or </>') may now be read from 
the reflection curves in Chapter 4. 

Equation (46) in Chapter 5 may now be applied 
with K = (F 2 /F 1 ) pD where D assumes the same 
values as in the approximate solution. Equation (21) 
determines the value of d from which u = d/dr 
may be calculated. This value of u = d/d T is laid 
off on the original divergence contour in Figures 4, 
5, or 6. This determines v. The assumption under¬ 
lying this procedure is that the divergence factor is 
not appreciably affected by the change in coordinates 
caused by imperfect reflection and an unsymmetrical 
antenna pattern. The corrected phase difference 8' 
is found from 


Basic Relations 

Referring to Figure 8 and assuming di « d 2 , 
y' < 10° and xf/ = y', the following relations hold. 



8 ' = ft - </>' 


(26) 


Figure 8. Lobe angles corrected for earth curvature. 


A' = — (— 


R' = 


kaA' 

hid? 


(1 

, , n\ 

tan y —> y = — 

4 hi' 

(29) 

(27) 

h' 


(28) Hence 

tan y' —» y' ~ - - . 

di 

(30) 


The intersections between the path-difference con¬ 
tours and the distance envelope determine points 
on the coverage diagram. 

The above method should be applied even for 
horizontal polarization when the directivity of the 
antenna is such that F 2 /F x ^ 1. This follows from 
the concept of generalized reflection coefficients of 
Section 5.3.1. 


and 


W 

di " 


n\ 

4V 


d _ (2fti0 2 

n\ 


where, from equation (58) in Chapter 5, 

d\ 


V = h- 


2 ka 


(31) 


The basic equations of the lobe-angle method are 


6 6 LOBE-ANGLE METHOD 

(HORIZONTAL POLARIZATION) 

6,6,1 Outline of Method 

In this method the angles of lobe maxima are 
determined by modifying the plane earth formula, 
equation (2). In this equation, hi is replaced by hi', 
which is the equivalent height above a plane tangent 
to the earth’s surface at the reflection point, as 
shown in Figure 8. 

The value of hi is given in equations (58) and (60) 
in Chapter 5. The maximum and minimum distances 
from the transmitter base to a point on the lobe 
are calculated by equation (46) in Chapter 5, using a 
modified divergence factor to be described in Section 
6.6.5. 



6 6 3 Reflection-Point Curves 

The elimination of di from equations (32) and (33) 
is most conveniently accomplished by graphical aids, 
which may be used in the following way. 

1. From equation (33), a curve may be plotted 
showing di as abscissa and n as ordinate for a given 
transmitter height and wavelength. This is illus- 







LOBE-ANGLE METHOD 


139 


trated in Figure 9 for two stations A and B with 
heights equal to 146.5 meters and 302 meters and 
wavelengths X = 1.50 meters and X = 1.42 meters 
respectively. 

2. From equation (58) in Chapter 5, a second curve 
may be plotted with di as abscissa and the equivalent 
height hi as ordinate as shown in Figure 10. To illus¬ 
trate, computed data for station A are given in 
Table 4, for a free-space range of d 0 = 100 km. d is 
calculated from equations (16) and (17). V 2ka, 


Table 4. Data for station A of Figure 9.* 


n 

d t 

(km) 

W 

(meters) 

7' 

(rad) 

Q 

(rad) 

7 = 7' ~ 

(rad) 

D 

Max. range 
d 

(km) 

0 

50.1 

0 

0 

0.00789 

- 0.00789 


0 

1 

28.0 

99.6 

0.00362 

0.00440 

- 0.00078 

0.502 

150.2 

2 

20.2 

122.5 

0.00590 

0.00317 

0.00270 

0.740 

26.0 

3 

15.9 

131.7 

0.00827 

0.00250 

0.00575 

0.817 

181.7 

4 

12.8 

137.0 

0.01058 

0.00200 

0.00858 

0.864 

13.6 

5 

10.9 

139.7 

0.01300 

0.00170 

0.00130 

0.910 

191.0 

6 

9.35 

141.4 

0.01540 

0.00145 

0.01395 

0.930 

7.0 

7 

8.05 

143.0 

0.01760 

0.00128 

0.01632 

0.943 

194.3 

8 

7.25 

143.6 

0.02000 

0.00112 

0.01888 

0.960 

4.0 

9 

6.45 

144.0 

0.02260 

0.00100 

0.02160 

0.964 

196.4 

10 

5.80 

144.2 

0.02510 

0.00090 

0.0242 

0.970 

3.0 

11 

5.32 

144.7 

0.0275 

0.00083 

0.02667 

0.973 

197.3 

12 

4.98 

145.0 

0.03000 

0.00078 

0.03022 

0.980 

2.0 

13 

4.51 

145.2 

0.03240 

0.00071 

0.03169 

0.982 

198.2 

14 

4.19 

145.7 

0.03480 

0.00066 

0.03414 

0.986 

1.4 

15 

3.87 

145.7 

0.03730 

0.00061 

0.03669 

0.989 

198.9 

16 

3.70 

145.7 

0.04000 

0.00058 

0.03942 

0.990 

1.0 

17 

3.48 

146 

0.04220 

0.00053 

0.04167 

0.991 

199.1 

18 

3.22 

146 

0.04450 

0.00050 

0.04400 

0.993 

0.7 

19 

3.06 

146.2 

0.04700 

0.00048 

0.04652 

0.994 

199.4 

20 

2.90 

146.2 

0.04960 

0.00045 

0.04915 

0.995 

0.5 


* Antenna gain and directivity factors have been omitted from the above calculations. 
tSee Section 6.6.4. 


where 


ka 


(35) 


Hence by equation (32) 


wX 


7 = 




di 
ka 9 


(36) 


3. For any n, including integral and fractional 
values, di may be found from Figure 9 and the 
corresponding hi from Figure 10. The angles y' 
corresponding to lobe maxima may then be calculated 
from equation (32). 

6 6 4 Lobe Angles with Horizontal 

The angle y' given by equation (32) is measured 
with respect to the tangent plane through the reflec¬ 
tion point shown in Figure 8. This plane is inclined 
at an angle 0 with the horizontal at the base of the 
transmitter. The true angle y which the lobe-center 
line makes with the hoiizontal is 

(34) 


where odd values of n give maxima and even values 
minima, provided the reflection phase shift is x 
radians. The angle may be either positive or nega¬ 
tive, as shown by equation (36). 

6 6 5 Use of Modified Divergence Factor 

The value of the divergence factor must be de¬ 
termined in order to calculate the maximum and 
minimum lobe lengths by equation (46) in Chapter 5. 
A convenient formula for the divergence factor at 
the angles of lobe maxima is obtained by substitut¬ 
ing y' for \f/ in equation (92) in Chapter 5. The 
errors involved in this assumption have been given 
in Section 6.2.3. Substituting y' =\f/in equation (92) 


7 = y' ~ d, 











140 


COVERAGE DIAGRAMS 


in Chapter 5 yields 


D = 


1 



2 V 
fca ( 7 ') 2 


For lobe maxima 


(37) 


Hence 



D = 


1 



n\ 

2 ka(y') z 


(38) 

(39) 



90 

80 

70 


10 

9 

8 

7 

6 

5 


w 

ID 

O 


.3 


.2 


Figure 9. Location of reflection point d\ as a function 
of lobe number n. 


by substituting K = (F 2 /Fi)pD and sin 2 (12/2) = 1. 
For horizontal polarization p = 1. Thus, 


dmax = — --Z)J +4 yD (40) 


or 


dm ax — *V&ido 






(41) 


Here F 2 andFi are computed from the angles^ and v 
given by equations (62) and (63) in Chapter 5. The dis¬ 
tance d min from the transmitter base to the minimum 

reflection point in kilometers Cstation b) 



Figure 10. Variation in effective .height hi with 
reflection point d\. 


Contours of constant y' are shown in Figures 11 point is obtained by substituting sin 2 (12/2) = 0 and 
and 12 where y f is a function of D and rih. K = ( F 2 /F\)D in equation (46) in Chapter 5. Thus 


Construction of Lobes 



(42) 


For horizontal polarization, the distance d max The values of D to be used in equations (40), (41), 
from the base of the transmitter to the lobe max- and (42) may be read directly from Figure 11 or 
imum is calculated from equation (46) in Chapter 5 Figure 12, or calculated by equation (39). 















































































LOBE-ANGLE METHOD 


141 


Intermediate points in the lobe may be formed by 
assigning fractional values to n. The corresponding 
path-difference angle 8 may be calculated in the 
following manner. Suppose it is desired to find 
intermediate points on the fourth lobe. The values 
of n for this lobe range from n = 6 to n = 8, with 
the maximum at n = 7. It follows that a change of 


as given in equation (46) in Chapter 5, in which 
K = ( Fz/Fi)pD and p = 1. The value of D may 
be read from Figures 11 and 12, using the assigned 
value of n\ and the relation y' = n\/4/q'. The 
proper value of di to be used in equation (63) in 
Chapter 5 to determine the antenna-pattern angle v 
may be read from Figure 9. 



Figure 11. Contours of 7' as a function of D and n\. 


2 in n corresponds to a change of 2r in 8. Thus if 
n = 6.5, 8 = (0.5/2) (2 t r) = tt/2 = 90°. For hori¬ 
zontal polarization ft ( = 8 + </>') reduces to 8, 
since (j)' = 0. The distance from the transmitter 
base to this intermediate point in the lobe is equal to 

d = Vftdo (1 - K)* + 4K sin 2 (|), (43) 


6 6 7 Correction for Low Angles 

The method outlined in Sections 6.6.2 to 6.6.6 
depends upon the assumption that y' —> \f/ or di —» 0. 
This assumption gives good results when n is a large 
number but serious errors are involved for small 
values of n. The method described in this paragraph 
is designed to avoid this difficulty. The procedure 































































142 


COVERAGE DIAGRAMS 


consists in plotting point by point the lobe-center Equations [(44a to e)] are obtained by inspection of 
lines and angles of lobe maxima. The points will be Figure 13. Equations (44f) and (44g) are derived as 
located by polar coordinates with the pole at the follows: 



Figure 12. Contours of y' as a function of D and nX. 


transmitter. The coordinates are shown in Figure 
13 as r d and 71 . 

Referring to Figure 13, the following relations 
hold when the angle \p is less than 10 degrees: 


sin 2\J/ 


r d 


sin (tp + 'I'd) B 

r d 


2\f/ 

7+77 


1 


(a) 

,_ h i_ hi' 

* A di 

(b) 

e = ^ = A 

ka ka 

(c) 

'<31 ’ 

1 

II 

£ 

(d) 

hi’ = hi - 

2 ka 

(e) 

I'd —> A + B = d\ + B 

(0 


(g) 

B _ (n\/2)di 

2 - n\/2 


(0 ^ (^r^) = * i^-~) ■ 

RECE.iVi.rt 



Figure 13. Geometry for radio propagation over 
spherical earth. See Figure 14 in Chapter 5. 



























































LOBE-ANGLE METHOD 


143 


The path difference A is given by 

nX 


A = A + B - r d = — (</> = 180°). 

2 


Hence 


nX 


A + B ~iJa 2 + B 2 + 2AB cos 2^ = 

-(?)'• 

-Kt? 


Squaring, 

R(2A — nX — 2A cos 2^) = nXA 
Therefore, 


R = 


nXA 


A(1 — cos 2\p) — 


nX 


For small angles, 


cos 2<p = 1 — ^ = 1 - 2\p 2 . 
2 


Hence 


n\A 

~2 


A 1 / nXV 
> ~~ 2 \~2 / 


nX 


. nx 
A — — 


2iA 2 A 


n\ 


2 . nX 

2A^ 2 — — 
2 


This method of determining the angles of lobe 
maxima is of particular value in constructing the 
first few lobes, since the approximations of Sections 
6.6.2 to 6.6.6 may cause considerable error for low 
angles. These path-difference loci will intersect a. 
vertical line drawn from the antenna to the ground 
below at heights equal to nX/4. For short waves, 
this height is negligible for the lower lobes. 


6 7 LOBE-ANGLE METHOD 

(VERTICAL POLARIZATION) 

6 71 Angles of Lobe Maxima 

The values of n corresponding to the angles of 
lobe maxima are determined exactly as in Section 
6.2.3 for the case of a plane earth. The values of n 
in the expression y' = nX/4/q' are increased above 
those for horizontal polarization by an amount (An) 
to compensate for the reduced phase shift on reflec¬ 
tion. In other words, the path difference must be 
greater than integral multiples of X/2 to compensate 
for the reduced phase shift. The expression for this 
compensation, from equation (5), is 

An = — (45) 

7r 


Since the angle \p is of the order of a few degrees 
only, it is permissible to write A = di in the above 
equation. 

Neglecting further the term nX/4 in comparison 
with di, which is permissible for short waves and 
small values of n, the above expression for B reduces 
to equation (44g). 

The method of determining the locus of points 
having a path difference of X/2 (i.e., n = 1) is as 
follows. Assume a value of di and calculate the 
corresponding values of 

hi' by equation (44d), 

\p by equation (44a), 

B by equation (44g), 
r d by equation (44e), 
i p d by equation (44f), 

0 by equation (44b), 

Yi by equation (44c). 

The assumed values of di are limited to those which 
will give positive values of B in equation (44g). 
Select as many values of di as are necessary to plot a 
smooth path-difference locus. Repeat for n — 3, etc. 


Hence 

y' = (n + An) . (46) 

Ahi 

[See Figure 8 and equation (32).] 

6 7 2 Construction of Lobes 

As a first approximation, the angles of lobe maxima 
are calculated on the basis of horizontal polarization. 
A table is constructed giving values of n and y' 
for maxima and minima. The next approximation 
is to let yp = y'. This assumes that di << d 2 . The 
values of 4>' and p may then be found from reflection 
curves, and (An) calculated from equation (45). 
The corrected values of y' may be determined from 
equation (46). It is simpler to find y' by interpolat¬ 
ing between integral values of n in the n versus y' 
table previously constructed. The values of d max 
and d min are 

d m ax = + K), (47) 

<2 mi „ = V(7 a d„(l - K), (48) 







144 


COVERAGE DIAGRAMS 


where K = (F 2 /Fi)pD. The divergence factor may 
be found directly from Figure 11 for the corrected 
values of n and 7 '. It will be found that for the higher 
lobes, the effect of (An) upon the value of the 
divergence factor is negligible. 

Table 5 shows calculations of the corrected values 
of n and 7 when the radiation from antenna A of 
6.6.3 and Table 4 is vertically polarized. Trans¬ 
mission over sea water is assumed. Tables 6 and 7 
illustrate the effects of vertical polarization on reduc¬ 
ing the maxima and increasing the minima. 


of n and minimum ranges. The free-space range is 
100 km, as in Section 6.6.3. The last column shows 
the ratio of ranges for horizontal and vertical polar¬ 
ization. 

68 GENERALIZED COVERAGE DIAGRAMS 
(HORIZONTAL POLARIZATION) 

6 81 Basic Parameters 

The u-v method applied to generalized coordi¬ 
nates which was given in Section 6.5.2 may be 


Table 5 


n 

cf)* 

(in degrees) 

</>'* 

(in degrees) 

An 


a 7 ' 

(rad) 

y' 

(rad) 


y = y' — 6 
(rad) 

0 

180.0 


0 

0 


0 

0 


- 0.00788 

1 

175.0 


5.0 

0.0278 


0.000064 

0.00369 


- 0.00071 

2 

171.5 


8.5 

0.0472 


0.000112 

0.00602 


0.00284 

3 

168.0 


12.0 

0.0664 


0.000155 

0.00843 


0.00593 

4 

164.7 


15.3 

0.085 


0.00204 

0.01080 


0.00878 

5 

160.6 


19.4 

0.108 


0.000250 

0.01325 


0.01153 

6 

157.3 


22.7 

0.126 


0.000290 

0.01569 


0.01423 

7 

153 


27.0 

0.15 


0.000374 

0.01807 


0.01681 

8 

149 


31.0 

0.172 


0.000413 

0.02061 


0.01947 

9 

144.5 


35.5 

0.197 


0.000491 

0.02309 


0.02208 

10 

140.0 


40.0 

0.222 


0.000532 

0.02563 


0.02472 

11 

135.2 


44.8 

0.249 


0.000623 

0.02812 


0.02735 

12 

130.5 


49.5 

0.274 


0.000685 

0.03068 


0.02991 

13 

125.8 


54.2 

0.301 


0.000722 

0.03312 


0.03241 

14 

120.8 


59.2 

0.329 


0.000790 

0.03559 


0.03493 

15 

116.0 


64.0 

0.355 


0.000886 

0.03819 


0.03778 

16 

110.2 


68.8 

0.388 


0.000968 

0.04077 


0.04019 

17 

105.3 


74.7 

0.415 


0.000995 

0.04320 


0.04177 

18 

101.1 


78.9 

0.437 


0.001091 

0.04569 


0.04519 

19 

96.1 


83.9 

0.466 


0.001130 

0.04823 


0.04775 

20 

91.8 


88.2 

0.490 


0.001224 

0.05082 


0.05037 

* <l> corresponds to y‘ in Table 4. <t>' 

— TV — 4>. 









Table 

6 




Table 7 




dmax (HP) 

dmax (VP) 

dmax (VP) 



dmin (HP) dmin (VP) 

dmin (VP) 

n 

K 

(km) 

(km) 

dmax (HP) 

n 

K 

(km) 

(km) 

dmin (HP) 

1 

0.904 

150.2 

145.5 

0.968 

2 

0.835 

26.0 

38.2 

1.47 

3 

0.765 

181.7 

162.5 

0.895 

4 

0.725 

13.6 

37.4 

2.75 

5 

0.670 

191.0 

161.0 

0.844 

6 

0.625 

7.0 

41.9 

5.98 

7 

0.585 

194.3 

155.2 

0.80 

8 

0.548 

4.0 

44.8 

11.2 

9 

0.516 

196.4 

149.8 

0.762 

10 

0.486 

3.0 

52.9 

17.6 

11 

0.458 

197.3 

144.6 

0.733 

12 

0.436 

2.0 

57.3 

28.6 

13 

0.415 

198.2 

140.8 

0.710 

14 

0.40 

1.40 

60.6 

43.3 

15 

0.385 

198.9 

138.0 

0.695 

16 

0.375 

1.0 

62.9 

62.9 

17 

0.362 

199.1 

135.7 

0.681 

18 

0.360 

0.70 

64.3 

91.9 

19 

0.360 

199.4 

135.8 

0.680 

20 

0.355 

0.50 

64.7 

129.4 


Tables 6 and 7 show the effect of a reflection 
coefficient which is less than unity upon the max¬ 
imum and minimum ranges of station A in Section 
6.6.3. The first table gives odd values of n and 
maximum ranges; the second table gives even values 


extended to all transmitter heights and wavelengths. 
In this method, points on the lobe are located by the 
intersection of the path-difference locus with the 
normalized distance envfelbpe. The basic parameters 
are do and R. 


















GENERALIZED COVERAGE DIAGRAMS 


145* 


In constructing a coverage diagram for a doublet 
transmitter, the transmitter height, the wavelength 
and the radio gain are known. It will be shown in 
Section 6.8.2 that the normalized free-space distance, 
dp = do/dx, is related to the gain factor A by the 
equation 

i=— d T A. (49) 

do 3X 


The path-difference parameter R has been expressed 
in equation (114) in Chapter 5 in terms of a height- 
wavelength factor r which is defined by 

R = nr, (50) 

where 


1 Ika X 

2 yl 2 h 3/2 ' 


(51) 


The first maximum, which for horizontal polariza¬ 
tion occurs when A = X/2, corresponds to n = 1, 
the second minimum to n = 2, etc. Recalling the 
discussion in Section 6.5.2, it follows that it is pos¬ 
sible to construct coverage diagrams in generalized 
coordinates with r the pattern or chart parameter 
and do the curve parameter on a chart for which r 
is fixed. 


6.8.2 


Determination of do 


It is possible to express dp = d 0 /d T in terms of 
E/E h the ratio of the field strength E corresponding 
to the lobe, to the free-space field Ei at unit distance 
from the transmitter. Since 
E i 


do = 


E 


it follows that 


do= do = 


(52) 

dj< dy E 

The ratio Ei/E may be expressed in terms of the 
gain factor A through the following relationships. 
By equation (16) in Chapter 2 
_ EW 
Pl ~ «• 


When d = 1, this gives 

Pi = 


El 

45* 


(53) 


For a doublet receiver with matched load and ad¬ 


justed for maximum power transfer, equation (17) 
in Chapter 2 gives 


E 2 3 1 
1207r St 


Hence 


Ei = 3X |Pi = 

~E 8t'P 2 


3X 
St A 


(54)* 


Substituting the value of Ei/E from equation (54) 
into equation (52): 

i-4(A), 


(55) 


and 


dy \StA/ 

i =rf K¥)z =v ^(lOi v ^ (56) 


Equation (56) shows that if log hi is plotted against 
log A for fixed values of do and X, a straight line- 
results. These straight lines are shown in Figure 14. 



Figure 14. Values of do as functions of hi and 20 log A 
—20 log X. (See equation 56. The letters refer to cover¬ 
age diagrams plotted in Figures 16 to 39.) 


If hi and E/E x or hi, X, and A are known, dp majr 
be calculated from equations (52) or (55). The* 
























































J4G 


COVERAGE DIAGRAMS 


X METERS 
.Olr- 

-02 — 

.03 |j- 

.041- 
.05 E- 

.1 L - 

. 2 '— 

.3|- 

4 =— 

.5 i- 

2E- 

3 h 

4 =— 

5r- 

.oC 


.001 

.002 

.005 

.01 

.02 



cr 

u 

co 

z 


cr 

< 

x 

o 


,ooo h ^ M£TERS 


500-E 

400-E 

300-E 


200 



Figure 15 


Chart number and r as a function of X and hi. (See Figures 16 and 17.) 
















GENERALIZED COVERAGE DIAGRAMS 


147 


value of do determines the range of the lobe for 
specified values of r. In Figure 14, the various 
values of do used in constructing the charts are 
specified as A, B, C, • • • , M, N, and are shown as 
functions of h as ordinate and 20 log A — 20 log X 
as abscissa. 

<,.8.3 Determination of r 

Figure 15 shows r for various values of transmitter 
height hi and wavelength X where 

1 = 2 Mt = hj^Zka = 2 /t! 3/2 IT 

r \ka \ka X ' ka 

The values of r determine how the path-difference 
curves intersect the envelopes corresponding to 
assigned values of sin 2 ( Q/ 2) in the equation 

v =4- \l(l - Z)) 2 + 4Z)sin 2 (58) 

a T > 2 

The generalized coverage charts are designated as 
1, 2, • • • , 11, 12 in Figures 16 to 39, with the chart 
number being given by Figure 15. 

6 8 . 4 Use of Charts 

Figures 16 to 39 give 24 basic charts developed 
by the Radiation Laboratory and presented in 
report 702. On each chart are complete lobes or 
lobe outlines labeled A, B, C, • • • , M, N. In the 
following description these letters are referred to as 
lobe letters and the numbers 1 to 12 as chart num¬ 
bers. It will be noted that each chart number is 
plotted to two scales, giving 24 charts in all. 

The problem of constructing coverage diagrams 
resolves itself into finding the chart number and lobe 
letter corresponding to given values of gain factor A, 
transmitter height h h and wavelength X. As stated 
in Section 6.8.1, the basic parameters of the general¬ 
ized coverage diagrams are R and do or r and do. 


The value of r is given in Figure 15 as a function of 
X and hi. Figure 15 was constructed from equation 
(51) which, after the substitution of numerical 
values, becomes: 

r= ^( f orfc = i). ( 59) 

The value of r determines the chart number. The 
lobe letter is found from the do corresponding to the 
given gain factor, transmitter height, and frequency. 
Figure 14 gives the lobe letters A, B, C, • • •, M, N 
as functions of 20 log A — 20 log X and the trans¬ 
mitter height. The relationship for plotting these 
lines is given by equation (56). 

As an illustration of the use of the generalized 
coverage diagrams, assume 20 log A = — 83, hi = 
33 meters, and / = 200 me (X = 1.5 meters). If a 
straight line is drawn connecting hi = 33 and X = 1.5 
in Figure 15, it will intersect the r scale at r — 8. 
Thus the chart number is 5. Now the lobe letter 
to be used in chart 5 must be found. For this case, 
20 log A — 20 log X = —86.48. The coordinates 
20 log A — 20 log X = —86.48, and hi = 33, de¬ 
termine the lobe letter to be E in Figure 14. Figure 
24 shows lobe E on chart 5. The first lobe is shown 
completely, together with the lower half of the second 
lobe. It must be noted that the coordinates of these 
charts., v = d/dy and u = h 2 /hi, are dimensionless. 
To convert to height h 2 and range d, the vertical 
distances must be multiplied by hi and the hori¬ 
zontal distances by d T . In this case hi = 33 meters 
and dy = V2fca/q = 23.6 km. The actual coordi¬ 
nates of the position of maximum range are h 2 = 
375 X 33 = 12.4 km and d = 15.4 X 23.6 = 365 km. 

The charts given in Figures 16 to 39 may be used 
for drawing coverage diagrams where the reflection 
coefficient is assumed equal to —1 and when the 
directivity factor F 2 /Fi is equal to unity. Each 
chart may be used for values of r near that for 
which the chart is drawn. For intermediate values, 
interpolation between charts is necessary. Errors 
inherent in interpolation limit the accuracy attained. 







148 


COVERAGE DIAGRAMS 



Figure 16. Generalized coverage diagram for r = 128. (See Figure 15.) 



Figure 17. Generalized coverage diagram for r = 128 (See Figure 15.) 



























































































































GENERALIZED COVERAGE DIAGRAMS 




Figure 18. Generalized coverage diagram for r = 64. (See Figure 15.) 



Figure 19. Generalized coverage diagram for r = 64. (See Figure 15.) 





























































































150 


COVERAGE DIAGRAMS 



CHART 3 

REFLECTION COEFFICIENT * — I 

CONSTANT GRADIENT OF REFRACTIVE INDEX 



v.-d- 


Figure 21. Generalized coverage diagram for r = 32. (See Figure 15.) 







































































































































GENERALIZED COVERAGE DIAGRAMS 


151 



Figure 22. Generalized coverage diagram for r = 1C. (See Figure 15.) 



Figure 23. Generalized coverage diagram for r = 16. (See Figure 15.) 




























































































152 


COVERAGE DIAGRAMS 



Figure 24. Generalized coverage diagram for r = 8. (See Figure 15.) 



Figure 25. Generalized coverage diagram for r = 8. (See Figure 15.) 



















































































































GENERALIZED COVERAGE DIAGRAMS 


153 



Figure 26. Generalized coverage diagram for r = 4. (See Figure 15.) 



Figure 27. Generalized coverage diagram for r = 4. (See Figure 15.) 



































































































154 


COVERAGE DIAGRAMS 




v = —— 
d T 


Figure 29. Generalized coverage diagram for r = 2. (See Figure 15.) 






































































































































GENERALIZED COVERAGE DIAGRAMS 


155 





Figure 30. Generalized coverage diagram for r = 1 . (See Figure 15.) 



Figure 31. Generalized coverage diagram for r = 1. (See Figure 15.) 































































































156 


COVERAGE DIAGRAMS 




Figure 33. Generalized coverage diagram for r = (See Figure 15.) 

















































































































GENERALIZED COVERAGE DIAGRAMS 


157 



Figure 34. Generalized coverage diagram for r = 34- (See Figure 15.) 



Figure 35. Generalized coverage diagram for r = . (See Figure 15.) 










































































































158 


COVERAGE DIAGRAMS 



Figure 36. Generalized coverage diagram for r = x /%. (See Figure 15.) 



Figure 37. Generalized coverage diagram for r = }/%. (See Figure 15.) 



















































































































































GENERALIZED COVERAGE DIAGRAMS 


159 » 



Figure 38. Generalized coverage diagram for r = lf 6 . (See Figure 15.) 



Figure 39. Generalized coverage diagram for r = \{§. (See Figure 15.) 







































































































Chapter 7 

PROPAGATION ASPECTS OF EQUIPMENT OPERATION 


71 GENERAL PROBLEM 

7,1,1 Introduction 

F or a standard atmosphere and with the basic 
assumptions set forth in Section 5.1.4, the 
relation between the factors affecting the power of a 
set and the gain factor A is given in equations (3) 
and (5) in Chapter 5. The problem of computing A 
depends on the set in the sense that some sets are 
designed to operate in free space, others with the aid 
of reflection from the sea, as in low-angle and surface 
coverage. 

The characteristics of a set as given in the manu¬ 
facturer’s description or in Tables 3, 4, and 5 at the 
end of this chapter may not represent the true values 
for a set in field use. Expected set performance, 
such as maximum range and coverage, can be calcu¬ 
lated on the basis of the set’s rated characteristics. 
Such performance can be termed “normal.” If a 
set is behaving abnormally, it may be that it is not 
functioning most efficiently. Unfortunately, the 
problem is complicated by the possible presence of 
atmospheric ducts and by the variability and diffi¬ 
culty of finding accurate^ the radar cross sections 
of aircraft and ships. 

Ducts are especially important for low antennas 
in surface search. In the case of communication 
sets, the most important item of information from a 
propagation standpoint is the maximum range. In 
the case of radar, not only is knowledge of maximum 
range wanted but also the ability to estimate the 
size and type of the target. 


7,1,2 The Performance Figure 

and Efficiency 

♦ 

The maximum range of a set depends on the peak 
power output P p of the transmitter, the minimum 
detectable power P min (see Sections 2.3.1 to 2.3.5) 
of the receiver, and the antenna gains Gi and G 2 . 
These can be grouped to give a performance figure. 
For communication, this figure is (P p /P m in)GiG 2 . 
For radar, the gains Gi and G 2 are generally equal. 
The performance figure is then ( P p /P m in)G 2 . The 
160 


ratio of the actual performance figure to the max¬ 
imum possible value, or the difference in decibels, 
gives the efficiency of the set. In field use, it is 
generally impossible to measure the working per¬ 
formance figure with any precision and methods for 
obtaining a rough measure must be employed. 


7,1,3 Effect of Reflection 

It has been pointed out in Chapters 5 and 6 that 
reflection may increase the maximum range of a 
radar up to twice the free-space value. This aid 





o Experimentoi Points 

Figure 1 . Effect of beam tilt on coverage for a radar. 

the early detection of aircraft at low angles. How¬ 
ever, the minima which occur in the resulting inter¬ 
ference pattern prevent the continuous tracking of 
an airship coming in. 























































GENERAL PROBLEM 


161 


This effect can be counteracted in several ways. 
One way is to employ microwaves whose inter¬ 
ference lobes are narrow and close together. Vertical 
polarization is another means of filling in the nulls 
while gaining in maximum range at low angles. 
Another device is to tilt the antenna beam upward 
so that some radiation (but substantially less than 
half) falls upon the sea. The result is a gain in low- 
angle coverage while the high-angle coverage is that 
of free space, without minima. 

The effect of various percentages of specular 
reflection in comparison with the free-space pattern 
is shown in Figure 1 for various beam tilts of a 
radar with a comparatively narrow beam width (11 
degrees between half-power points). The experimen¬ 
tal data, shown by small circles, illustrate the increase 
in detection range at 0 degree while ^t 5 degrees 
elevation angle there is little gain over free space. 

The roll of a ship, by varying the beam tilt, results 
in a shift in coverage, as can be seen from Figure 1. 


7,1,4 Signal-to-Noise Ratio 

The visibility of a signal on a scope depends on its 
relation to the noise. In early work with radar, the 
maximum range was defined by a ratio of signal 
voltage S to noise voltage N of unity, i.e., 

* = i. (i) 

N 

However, as pointed out in Section 2.3.5, the min¬ 
imum detectable signal is greater than N. Since the 
pip on a scope includes noise, equation (1) is equiva¬ 
lent to 


S + N 
N 


( 2 ) 


On an A scope, this relation signifies that the height 
of the signal is twice that of the noise grass. 

Since the visual signal in a set functioning properly 
varies linearly with the signal voltage, the size of 
targets can be estimated by means of the size of the 
visual signal. The ratio S/N gives a means of meas¬ 
uring a signal in terms of the noise. To change 
(S + N)/N to S/N, the value of (S + N)/N is 
expressed with unity as denominator. For example, 
if (S + N)/N is estimated to be 8/2 from the scope, 
the equivalent fraction is 4/1. The value of S/N is 
(4 - 1)/1 = 3/1. 


The relation of receiver power, P 2 , to noise power, 
NP, and the signal-to-noise ratio, S/N, is given by 

10 log —i = 20 log — . (3) 

NP 6 N 

7,1,5 Calibration of an A Scope 

On an A scope, the ratio (S + N)/N can be 
estimated roughly by eye. To improve upon this, a 
calibration is employed. One method is to mark the 
A scope to facilitate the reading of heights. Another 
method goes beyond this and calibrates the gain 
control. A turn of 5 db is equivalent to a raito of 
1.8/1. A datum line 1 cm above the time line and 



Figure 2. Method of calibrating an A scope. 



Figure 3. Calibration of gain control of an A scope. 

(Dotted lines are uncorrected calibration.) 

another at 1.8 cm are drawn on the A scope. The 
noise is brought up to the datum line by means of 
the gain control. The position is marked 0 on the 
gain control (see Figure 2). A steady signal is found 
(permanent echo, large boat, or signal generator) 
which produces a signal height of 1.8 cm above the 
time line. The gain control is then turned until the 
signal is reduced to the datum line. The position 









162 


PROPAGATION ASPECTS OF EQUIPMENT OPERATION 


on the gain control is marked 5 db (see dotted lines 
in Figure 3). 

Keeping the 5-db position on the gain control, 
another signal is obtained which comes up to 
1.8-cm line. The gain control is turned until the 
signal is brought down to the datum line. The new 
setting is marked 10 db. This is repeated until 
markings up to about 70 db are obtained. 

This calibration of (S + N)/N must be corrected 
to S/N (Figure 3) which can be done by means of 
Table 1. For 25 db and above, the values of 
(S + N)/N can be taken as equal to S/N. Any 
signal voltage can then be measured in decibels 
above the noise voltage ( N ) by turning the gain 
control until the signal height is 1 cm. By equation 
(3) this is also the signal power received, in decibels, 
above the noise power which is discussed in Sections 
2.3.1 and 2.3.2. 


Table 1 . Correction of (S + N)/N to S/N. 



Corrected 

o 

Uncorrected 

( s r) a » 


0 

6 


5 

9 


10 

12.5 


15 

16.5 


20 

21 


In the calibration just described, the pip on the 
scope is supposed to be proportional to the received 
signal strength. In a set functioning normally, this 
is justified, but occasionally defects in the set may 
destroy the linearity. The existence of a linear rela¬ 
tion can be tested by means of a signal generator. 

72 FREE SPACE — HIGH-ANGLE 

COVERAGE 

7,2,1 Maximum Range Formulas 


If P 2 is replaced by the minimum detectable power 
of the receiver, P min , and P 2 by the peak power P p 
of the transmitter, the maximum range is given by: 


One-way, 


Two-way, 


-E\ 




G 2 \ 2 


9(7 


( 6 ) 

(7) 


7 2 2 Deviation from Maximum of Beam 

For an antenna whose direction is fixed, the 
equations in Section 7.2.1 apply only to points on 
the axis of the beam. Denoting by/(r) the ratio of 
gain in a direction at an angle r from the axis of the 
beam to the gain at the axis and by 2 r 0 the beam 
width between half-power points, then 

/( r ) = exp - 0.692(r/ to) 2 . (8) 

Accordingly, for points off the axis, G must be 
multiplied by/(r) before substitution in the formulas 
of Section 7.2.1. 



In free space, the gain factor A has the value 
3X/87T d (for maximum power transfer between 
doublets). Equations (3) and (5) in Chapter 5 
then take the simple forms: 

One-way, radio gain: 



Two-way, radar gain: 

= 1( P r Z ( _3X V 

Pi 9A 2 \8tt d ) ’ 


Figure 4. Scanning loss as a function of scanning 
speed and beamwidth. 

7 2 3 Performance Figure 

Equations (6) and (7) for d max depend on the 
performance figure defined in Section 7.1.2. The 
quantities which appear in the performance figure 
can be measured. The one which offers most diffi¬ 
culty is P m in, the minimum detectable power, which 
has been discussed in Section 2.3. In Tables 3 and 4 at 

































LOW-ANGLE AND SURFACE COVERAGE 


163 


the end of this chapter, noise figures and band widths 
of various sets are given. An important correction to 
P m i., as determined from the noise figure and band¬ 
width is the scanning loss. This loss for various 
scanning speeds and beamwidths is represented in 
Figure 4. Another source of loss is deviation of the 
product of bandwidth B (me) and the pulse width 
t (microseconds) from the optimum value of 1.2. 
The losses for various values of the product are tabu¬ 
lated in Table 2. 


Table 2. Loss resulting from band- and pulse widths. 


Bt * 

Loss (db) 

0.1 

5.0 

0.3 

1.5 

0.7 

0.5 

1.2 

0.0 

2.5 

0.8 

5.0 

3.0 

10.0 

5.0 

20.0 

8.0 


*B = i-f bandwidth (me); t = pulse width (microseconds). 


A field measure of the performance figure of a 
radar can be determined by the use of a target of 
known radar cross section, such as a silvered balloon 
and equation (7). A check on variability of per¬ 
formance can be made by finding the maximum 
range on a plane (using a constant aspect, such as 
nose or tail). 

7 2 4 Radar Cross Section 

An important but troublesome factor in calculat¬ 
ing d max of a radar from a knowledge of the perform¬ 
ance figure is a, the radar cross section (see Section 
2.4.1 and Chapter 9).The value of o-can be found by: 

1. Laboratory measurement of the factors which 
constitute the performance figure and field determi¬ 
nation of the maximum range d max . The value of 
a is then given by equation (7). 

2. Measuring the signal returned by the target 
at a convenient distance on a calibrated A scope or 
by direct comparison with a pulsed signal generator. 
In this method neither P min nor d max enter. 

The equations involving <r assume a point target. 
Since an airplane intercepts a small solid angle over 
which the beam strength varies little, the assumption 
of a point target is adequate for aircraft. 


7 3 LOW-ANGLE AND SURFACE 
COVERAGE 

7 31 Maximum Range 

Since the gain factor A for this case is more 
complicated than for free space, the relation be¬ 
tween d max and the performance figure cannot be 
given in general by a simple expression, as can be 
seen from equations (172) and (184) in Chapter 5. 
For ranges such that the shadow factor F s ~ 1, i.e., 
distances d less than 10 4 \ 1/3 , X, and d in meters, and 
both antennas low, or antenna and target low (hi,h 2 < 
30\ 2/3 ), the form of A is simplified so that a simple 
relation can be given for d max . Otherwise, the 
methods of Sections 5.6, 5.7.3, and 5.7.4 must be 
employed, especially of Sections 5.6.5 and 5.7.3. 

73 2 Ducts and Set Performance 

It has been found from field tests that atmospheric 
ducts are likely to be found close to the surface of the 
sea. The consequent increase in range may mask 
subnormal set performance. If the antenna is 
tilted upward so that no radiation reaches the earth, 
then the free-space discussion of Section 7.2 applies 
to the field determination or check of the per¬ 
formance figure. Otherwise field testing, when condi¬ 
tions are normal, can be accomplished by the use of 
permanent echoes or the use of a ship of known 
target cross section (see Section 7.3.7). 

7 3 3 Low Heights and Plane Earth Ranges 

For these conditions, the following relations must 
hold: 

(h m < 30X 2 3 , d < 10 4 X 1/3 ), 

where h, d and X are in meters (see Section 5.7.1). 

In the dielectric case (see Section 5.7.3), generally 
applicable to radar, the value of A [equation (172) 
in Chapter 5] for F s = 1 and g = 1, becomes 


Since A = A 0 A P and A 0 = 3X/87T d, the preceding 
equation is equivalent to a path-gain factor value of 

Ap = ( 10 ) 

\d 

[See equation (55) in Chapter 5.] Instead of equa¬ 
tions (4) and (5), we now have [from equations (3) 
and (5) in Chapter 5] for a dielectric earth, 







164 


PROPAGATION ASPECTS OF EQUIPMENT OPERATION 


One-way, radio gain: 


P% 

Pi 


— —G1G2 

4 


Wh.2 2 

d 4 


Two-way, radar gain: 


p% 

Pi 


= 9t rG» 


\ 2 d 8 


the curve parameter instead of A. Equation (5), 
Chapter 5, can be written 

( 11 ) 

20 log A = — 5 log G 2 — 5 log <7 — 5 log 

P min 0X“ 

(12) (15) 


Hence for maximum range, replacing P x 
and P 2 by P m j n , 

by P p , 

One-way, 

4., = J- W ( C n G ,) . 

4 Vmin ' 

(13) 

Two-way, 

' x 2 \P m J 

(14) 

The maximum range now depends on the antenna 
heights and on the target height. 

7 . 3.4 

Radar Cross Section 
of Surface Craft 



Effective Height of Targets. Since the field varies 
considerably with height for a low target, the 
scattering by even simple geometrical objects 
requires integration. The formulas for radar in 
Section 7.3.3 apply to a point target with radar 
cross section a at an effective height h efi . 

In the case of a cylinder of radius a and length 
H, the radar cross section in a uniform field is 
a = 2 tt aH 2 /\. Under operating conditions the field 
is not uniform but a feasible approximation may be 
obtained by using the preceding value of a and a 
value of h% equal to the average value of the target 
height. 

The radar cross section of a ship is a complicated 
problem which is not discussed here. 



Figure 5. Maximum range versus height curves for 
various targets. 


7,3,5 Maximum Range 

Versus Height Curves 

By the method of Section 5.6.6 (see also Sections 
6.5.2 and 6.8.4), maximum range versus height 
curves can be constructed for various values of A. 
These are of importance chiefly for low-angle aircraft 
coverage. If the performance figure, (Pp/PmiJG 2 , 
is known, equation (5), in Chapter 5, defines the 
relation between <7 and A so that a can be taken as 


A set of curves for a 10-cm radar is given in 
Figure 5 for a transmitter height of 30 meters. The 
curve corresponding to a = 31 square meters is indi¬ 
cated. The corresponding value of 20 log A = —130 
was found from equation (15), using the value of 
218.4 db as the performance figure. 

For a ship with an effective height of 18 meters, 
Figure 5 gives the value of 20 log A at various dis¬ 
tances. With the aid of equation (5), in Chapter 5, 
the power P 2 returned to the radar by the ship at 
these distances can be found, using the following 


























































































Height-* Meters 


LOW-ANGLE AND SURFACE COVERAGE 


165 



Figure 6 . Maximum range versus height curves for various targets. 















































































































































166 


PROPAGATION ASPECTS OF EQUIPMENT OPERATION 


data: P x = P p = 60 kw, G = 22.5 db, and <r = 7,440 
square meters. 

In Figure 6, the maximum range versus distance 
curves for a set are presented. This set has a beam- 
width of 60 degrees so that it is effective for tracking 
aircraft coming in toward the set (increasing angle 
of elevation). The near coverage then approximates 
the free-space coverage which is dependent on the 
angle of inclination of the antenna. 

In Figure 7, curves are presented for a 3-cm set. 



Figure 7. Maximum range versus height curves for 
various targets. 


Estimating Ship Size 

The fact that the strength of a signal received 
from a target depends on the size of the target can 
be utilized to estimate ship size. If signal strength 


versus distance curves are obtained for surface craft 
of various sizes, then by plotting readings from an 
unknown vessel, its size may be estimated. A field 
procedure for obtaining these curves is to use a 
number of surface craft of assorted sizes and follow 
them on a radar with a calibrated A scope, recording 
signal strength versus distance. If a number of such 
runs are taken during periods judged to be standard 
and the results averaged, a fairly reliable chart 
should result. 

The curves can be obtained by calculation using 
measured power output and antenna gain and 
empirically determined values of <r in equation (5) 
in Chapter 5. The shape of such a curve for a “point” 
target is given in Figure 26 in Chapter 5. The radar 
curve can be obtained from this curve by changing 
the ordinate back to 20 log A [equation (3) in 
Chapter 5] and then using equation (5) in Chapter 5. 

Since an actual surface target is generally large 
and at short distances intercepts a considerable 
part of the energy, the oscillatory part of the point 
curve of Figure 26 in Chapter 5 is smoothed out. 
The value of A for short distances is sometimes 
taken as 6A 0 , or 15 db above the free-space value. 

In Figure 8, an example is given of curves com¬ 
puted for various ship sizes. Most of the plotted 
points fall in the 200- to 600-ton region. However, 
the precaution of a performance check is necessary 
in order to use this method most effectively (see 
Section 7.3.7). Since shape plays an important part 
in radar cross section, a ship of unusual shape may 
give a signal outside its class. 


7 3 7 Performance Cheek 

Since the performance of a set varies, before a 
chart described in Section 7.3.6 is used to detect 
unknown ships, it should be checked to determine 
whether set performance is normal. The curves 
can be used for this check by using a known ship. 
Such a check is shown in Figure 9 for two different 
days. On one day, the points for the ship fell in 
the correct region. On the other day, they fell too 
low by about 20 db. Accordingly, on the latter day 
the curves should have been used with an ordinate 
correction of —20 db. 

Other types of performance check may be used, 
such as permanent echoes, free-space range of air¬ 
craft, or a signal generator. 










































































































LOW-ANGLE AND SURFACE COVERAGE 


167 



Figure 8. Estimation of ship sizes. (From Coast Artillery Experimental Establishment, England.) 
































168 


PROPAGATION ASPECTS OF EQUIPMENT OPERATION 



Figure 9. Range in thousands of meters. (From Coast Artillery Experimental Establishment, England.) 









































DATA ON COMMUNICATION AND RADAR EQUIPMENT 


169 


DATA ON COMMUNICATION to be exhaustive. These are the best available data 

AND RADAR EQUIPMENT but ghoyjd be used with caution, since specification 

The tables given in this section are not intended changes may change rated characteristics of sets. 


Table 3. Communication equipment. 


Communication 

equipment 

Frequency 

(me) 

Power 

output 

(watts) 

Receiver 

sensitivity 

Antenna 

Polarization 

Gain 

(db) 

Beamwidth 

Horizontal Vertical 

SCR-608 

30 

25 

0.2 /xv 

V 


Whip 

SCR-300 

44 

0.5 

2.5 /xv 

V 


Whip 

AN/TRC-1 

70-100 

50 

25 /xv 
(4 channel) 

H 

6 


SCR-522 

125 

6 

4 /xv 

V 


Whip 

AN/TRC-8 

240 

12 

30 /xv 
(4 channel) 

H 

7 


AN/TRC-5 

1,400 

400 

NF* 12 db 

H 

14 


AN/TRC-6 

4,600 

2 

NF* 19 db 

H 

33 


TBS 

70 

50 

5 /xv 

V 

1 


AN/ARC-1 

125 

10 

5 /xv 

V 


Whip 


* Noise figure. 


Table 4. Radars. 


Receiver 

Power sensitivity 


Radars 

Frequency 

(me) 

output 

(kw) 

Noise figure 
(db) 

Bandwidth 

(me) 

Antenna 

Polarization Gain (db) 

Beamwidth 
Horizontal Vertical 

SCR-271DA 

106 

100 

6 

1 

H 

19.8 

11° 

12° 

AN/TPS-3 

600 

200 

11 

1.8 

H 

23.4 

12° 

11° 

SCR-584 

3,000 

300 

15 

1.7 

H 

30.8 

7° 

7° 

AN/MPG-1 

10,000 

60 

17 

10 


40.8 

0.6° 

3° 

SC-4 

200 

200 

6 

0.5 

H 

13.5 

20° 

60° 

SQ 

2,500 

1 

13 

2 

V 

20 

8° 

15° 

SG-1 

3,000 

50 

18 

1 

H 

28 

6° 

15° 

ASB 

515 

10 

15 

1.4 

H 

30 

o 

O 

CO 



Table 5. IFF or beacons. 


IFF or beacons 

Frequency 

(me) 

Power 

output 

Receiver 

sensitivity 

Antenna 

Polarization 

Gain 

(db) 


AN/TPX-4 

170 

0.5 kw 

15 Mv 

V 

6 


AN/UPN-1,2 

3,000 

50 kw 

0.05 Mv 

H 

6 


AN /UPN-3,4 

9,320 

300 kw 

0.02 /xv 

H 

12 
















Chapter 8 

DIFFRACTION BY TERRAIN 


81 OUTLINE OF THEORY 

811 Introduction 

T he effects of diffraction around natural ob¬ 
stacles of complicated shape are difficult to 
analyze. Theory offers two lines of approach to 
diffraction problems, both based on the substitution 
of contours of simple shape in place of the natural 
obstacle. 

The first and oldest method, known as the Fresnel- 
Kirchhoff method, is an approximate procedure for 
calculating the diffraction by a flat screen. It yields 
comparatively simple formulas for the diffracted 
field; the present chapter is concerned with a 
presentation of this method. 

The second method is based on the fact that the 
wave equation can be solved for obstacles of very 
simple geometrical shape, especially cylinders and 
spheres. If the curvature of a hill is fairly constant, 
so that its shape can be approximated by a cylinder 
or sphere, the field behind the hill can be obtained 
by the use of this method. 

Observations on diffraction by obstacles in the 
short wave and microwave region are very sparse. 
It is, therefore, not possible to present a consistent 
body of results that could be utilized in radio prac¬ 
tice. It seems, however, rather certain from the 
observations that when the shape of the obstacle 
approaches one of the special shapes dealt with by 
the theory, the latter gives a fair account of the 
facts. Such cases will not be found too frequently 
in practice. The hope is nevertheless justified that 
the right order of magnitude is obtained by a judi¬ 
cious application of the theory. The main applica¬ 
tion is in the lower frequency band (30 to 200 me); 
for higher frequencies, the diffracted field is rela¬ 
tively unimportant. 

8,1,2 The Fresnel Diffraction Theory 

The Fresnel-Kirchhoff approximate theory was 
originally developed to account for the diffraction of 
beams of light when cut off by diaphragms, slits, and 
similar optical devices. In applying this theory to 
170 


the propagation of radio waves over the earth, only 
one basic problem is usually encountered, namely 
that of diffraction around a straight edge. In the 
present section, the general method of handling this 
problem and obtaining numerical results is given. 
On applying the method to actual cases certain 
accessory problems arise which will be dealt with 
in Sections 8.3 and 8.4. The most important of these 
complications is caused by ground reflection. 



Figure 1 . Diffraction around straight edge. 


In Figure 1, the area CAPBD forms an opaque 
screen bounded by a straight edge BP A. The width 
AB of the screen is assumed infinite in the mathe¬ 
matical theory, but is here shown finite for simplicity. 
The line connecting the transmitter T to the re¬ 
ceiver R intersects the plane of the opaque screen 
in the point M whose distance from the edge is 
PM = h 0 . The shortest unobstructed path of the 
radiation is TPR. 

In a purely geometrical theory, the point R would 
be in the shadow of the screen and would receive no 
radiation. If the wave nature of radiation is taken 
into account, it is found that an electromagnetic 
field is generated in the shadow of the screen; the 
waves are bent around the obstacle. 

The mathematical derivation of the diffraction 
formulas will not be given here as it is rather intri¬ 
cate; however, the problem is discussed in Section 
8.2. The discussion here is limited to a qualitative 
visualization of the mechanics of diffraction (Sec¬ 
tion 8.1.3); the final formulas used for computa- 







OUTLINE OF THEORY 


171 


tions will then be written down at once (Sections 
8.1.4 to 8.1.6). 

813 Mechanism of Diffraction 

The physical idea underlying the Fresnel-Kirch- 
hoff diffraction theory may be presented as follows. 
At points visible from T the field, to a first approx¬ 
imation, is equal to the free-space field E 0 . This 
applies in particular to all points of the plane ECDF 
containing the screen. The receiver R receives 
radiation from the open part EAPBF of the plane, 
while there is no radiation incident upon R from the 
opaque surface CAPBD. In order to compute the 
field at R, it is assumed that in the open part of the 
plane the field is E 0 while on the opaque screen the 
field vanishes. Such a field distribution may be 
realized physically by assuming that there is a con¬ 
ducting sheet in the open region EAPBF with 
suitably chosen oscillating charges or currents such 
that the field E 0 is produced on the side of the sheet 
facing the receiver. The total radiation received 
at R from such a current-sheet will be equivalent 
to the radiation from T bent around the diffracting 
screen. The fictitious sheet EAPBF forms a system 
of secondary sources of radiation whose effect is 
equivalent to that of the primary source for all 
points on the far side of the plane ECDF (side of the 
receiver), but not on the near side (side of the 
transmitter). 

It is evident that most of the radiation received 
at R comes from the area near the point P above the 
line APB. The relative importance of contributions 
of areas more or less removed from P is discussed in 
Section 8.2. 

When the primary source at T is replaced by a 
distribution of secondary sources in the plane of the 
screen, an essential approximation is made. It is 
assumed that there are no secondary sources in the 
opaque region CAPBD. In reality, the screen is a 
physical body and, whether it is a conductor or a 
dielectric, there is an electromagnetic field in its 
surface layers, especially near the edge APB , and 
this field makes a contribution to the radiation 
received at R. In the approximate theory, it is 
assumed that the field on the surface of the opaque 
screen is negligible. In the terminology of optics, 
this implies that the screen is black ; in radio termi¬ 
nology, it means that the surface of the screen is 
rough (Section 8.3.2). 


814 The Straight Edge Formula 

The physical picture just described can be put 
into mathematical language. When the rather 
intricate derivations are carried through, a rela¬ 
tively simple formula results. 

The symbols and designations used are illustrated 
in Figure 2. In accordance with practice it is 
assumed that the line TR is nearly horizontal. The 
trace of the opaque screen on a vertical plane through 
T and R is assumed perpendicular to the fine TR 



\» 

Figure 2. Diffraction around straight edge. 


(upper part of Figure 2). The trace of the screen 
on a horizontal plane may, however, make an 
angle </> with the line TR (lower part of Figure 2). 

In view of the approximate nature of the theory 
explained in the preceding paragraph, the following 
conditions must be fulfilled in order to obtain 
reliable results: 

d 1 ,d 2 >>h 0 >>\. (1) 

That is, the distances from the transmitter and 
receiver to the obstacle must be large compared to 
the height of the latter above the line TR, and this 
height must be large compared to the wavelength. 
The second of these conditions is likely to be ful¬ 
filled in the short-wave and microwave bands, and 
the first will be fulfilled when the angles of elevation 
ai and a 2 of the rays, drawn from the transmitter 
and receiver to the edge, are small. 

A second condition for the validity of the diffrac¬ 
tion formula refers to the horizontal extension of the 
screen. The formulas are derived for a screen of 
infinite horizontal extent, but in practice it will 
usually suffice if the horizontal extension of the 
screen is large compared to the height h 0 . 








172 


DIFFRACTION BY TERRAIN 


If these conditions are fulfilled, the field at the 
receiver is given by 

E = E °^J-J in,/2dv ’ (2) 

where Eo is the free-space field at the receiver in 
absence of the screen and 

(3) 

In the last formula, use is made of the fact that 
ai and a 2 are small angles so that approximately 
«i = Ao/dianda 2 = ho/d^. 

The Fresnel Integrals 

An integral of the type appearing in equation (2) 
is known as Fresnel’s integral; its properties will 
now be briefly discussed and numerical data given. 
The standard Fresnel integral is usually defined as 

C(v) - jS(v) = j\<-~ *' 2 » dv, (4) 


where 



S(y) — J sin v^j dv. 


If this function is plotted in the complex plane, 
with C and S as abscissa and ordinate, respectively, 
for all values of v, a curve is obtained that is known 
as Cornu’s spiral (Figure 3). C — jS is represented, 
in magnitude and phase, by a vector from the origin 
to a point on this spiral. 

It may be shown that the length of arc along the 
spiral, measured from the origin, is equal to v. 
In the graph, values of v, counted positive in the first 
quadrant and negative in the third quadrant, are 
indicated along the spiral. As v approaches infinity, 
the spiral winds an infinity of times around two 
points lying at the distance 1/V2 from the origin 
on a 45-degree line. C and S for the end points are 
iv i 1 

C(droo) = ± —, $ (± o°) = ± - (5) 

2 ^ 



Figure 3. Cornu spiral. 












































OUTLINE OF THEORY 


173 


SA b Application to Straight Edge 

Since 

' = l+l 

V2 2 ’ 

equation (2) may be rewritten, on using equations (4) 
and (5), as 

E = E ° ~ [l + C(v) - j - jS(»)] . (6) 

It will be noticed that the quantities 

i + C '4 + ' S ) 

have a simple geometrical meaning. They are the 
real and imaginary components, respectively, of a 
vector drawn from the lower point of convergence 
(point —1/2, —i/2) of the Cornu spiral to a point 
on this spiral. The bracket in equation (6) is equal 
to this vector in magnitude but with opposite phase. 

The Fresnel formulas and Cornu spiral as given 
above will assist the reader in establishing the rela¬ 
tions of our equations with the classical theory of 
diffraction as found in all textbooks on the subject. 
For practical purposes the field behind a diffracting 
straight edge given by equation (6) will be denoted by 


In Figure 4, the modulus s is plotted as a function of 
v. In Figure 5, the phase lag f is plotted in a similar 
way. (With the above choice of the sign, f is positive 
in the shadow.) 

The variable v is given by equation (3). On 
account of the square root, there is an ambiguity in 
sign. Closer inspection shows that v must be taken 
positive when the receiver is in the illuminated region , 
above the line of sight; v must be taken negative when 
the rec eiver is in the shadow zone. 

When v tends to - oo , the line APB (Figure 1) 
moves far upwards relative to the line TR; the 
receiver lies deep in the shadow and E approaches 
zero by equation (6). When v tends to + oo , the line 
APB moves far downward, and the screen ceases 
to orm an obstruction, E approaches E 0 . At the line 
of sight (when the point P in Figure 1 coincides 
with M), v = F{v) = 0 and E = E 0 / 2 . Clearly, the 
effects of diffraction are not confined to the shadow 
region but extend considerably into the illuminated 


zone. If the receiver is sufficiently deep in the 
shadow, about v > — 1 , the following approximate 
formula holds: 



0225 

v 



0 -I -2 -3 -4 -5 


Figure 4. Magnitude of relative field strength E/E 0 
versus v. 

































174 


DIFFRACTION BY TERRAIN 


81,7 Polarization. Large Angles 

It may be noticed that in the preceding equations 
no reference is made to the state of polarization of 
the diffraction field. The results of the approximate 
Fresnel-Kirchhoff theory are independent of the 
state of polarization in agreement with observation. 



12 ° 



- 18 ° 


ABOVE LINE OF SIGHT V-► 

Figure 5. Phase lag (ordinate) of relative field 
strength (E/E 0 ) versus v (abscissa). 

If the angles of diffraction (<* x and a 2 , Figure 2) 
become large, larger than a few degrees, for instance, 
the approximate theory no longer applies. The 
deviations from the Fresnel formulas then go in 
opposite directions for the two states of polarization. 
If the electric vector is parallel to the diffracting 
edge, the field in the shadow at large angles is 


slightly diminished as compared with that given by 
the Fresnel formulas; if the electric field is per¬ 
pendicular to the diffracting edge, the diffracted 
field in the shadow at large angles can become appre¬ 
ciably larger than the calculated one and, in the case 
of very large angles, the excess may reach the mag¬ 
nitude of, say, 6 to 15 db. In the region above the 
line of sight, the sign of the polarization effect is 
reversed (slight increase for polarization parallel to 
the edge, appreciable decrease for polarization 
perpendicular to the edge). 

These effects are entirely analogous to those that 
are observed when the currents induced in the surface 
of the obstacle cannot be neglected (Section 8.1.3), 
and they have the same physical origin. 

82 DIGRESSION ON FRESNEL’S THEORY 
821 Fresnel Zones 

The concept of the Fresnel zone has played an 
important role in the development of diffraction 
theory. As it is frequently referred to in papers on 
the subject, it may be useful to digress briefly on it. 
Fresnel’s original construction is based on the con¬ 
ception that any small element of space in the path 
of a wave may be considered as the source of a 
secondary wavelet, and that the radiation field can 



Figure 6. Relations of Fresnel zones and diffracting 
slots. 


be built up by the superposition of all these wave¬ 
lets (Huyghens’ principle). In particular, consider 
the field produced by the transmitter in the open 
part of the plane containing the diffracting screen 
(EAPBF in Figure 1) and let each element of this 
plane be the source of a secondary wavelet. This 
may be achieved by distributing a suitable ficti- 


































DIFFRACTION BY HILLS 


175 


tious system of oscillating currents (or a system of 
elementary doublets of proper strength) over the 
surface of the plane. The field at the receiver is then 
the superposition of all the fields produced by the 
wavelets. 

Now let (Figure 6) S be a plane perpendicular to 
the line TR and let M be the point in which the line 
TR intersects the plane S. Let Q be a point on the 
plane S such that the difference in path between 
TQR and TMR is just X/2. The locus of these 
points is a circle about M. Similarly we can con¬ 
struct other circles so that the corresponding path 
differences are integral multiples of X/2. The area 
within the first circle is called the first Fresnel zone, 
the subsequent ring-shaped areas are called the 
second, third, etc., Fresnel zones. The secondary 
wavelets originating in the first, third, fifth, etc., 
Fresnel zones are in phase with each other and rein¬ 
force each other by constructive interference at R, 
while the secondary wavelets originating in the 
second, fourth, etc., zones are in phase with each 
other but out of phase with the former group and 
tend to cancel the field produced by this group. 

Hence if the plane S is opaque except for a round 
hole centered on M, the intensity of the radiation 
field at R will depend on the number of Fresnel 
zones that fall inside the hole. If we start out with a 
very small hole and progressively increase its size, 
there will be a maximum of intensity at R (nearly 
twice the free-space field E 0 ) when the hole just 
comprises the first Fresnel zone. If the size of the 
hole is further increased, the destructive interference 
of the second zone comes into play, decreasing the 
intensity, and a minimum (very nearly zero) is 
reached when the hole contains just the first two 
zones. On continued increase of the hole size, 
further maxima and minima appear. The amplitude 
of these oscillations decreases very gradually until 
eventually the field at R approaches the free-space 
value. 

822 Diffraction by a Slot 

The preceding considerations indicate that only a 
comparatively small area of an opening, of the order 
of one Fresnel zone, is required to produce an 
illumination that is comparable in order of magni¬ 
tude to the free-space field. It is also seen that the 
simple geometrical construction of the Fresnel zones 
is more suitable when dealing with the diffraction 


by round openings than with screens bounded by 
straightedges. Qualitatively, however, the conditions 
are similar. 

As an example, consider the case of a slot bounded 
by parallel edges at distances h 0 and h 0 ' from the 
point of intersection M between the plane of the 
slot and the direction from the observer to the 
distant light source (see Figure 6). The diffracted 
field E will obviously be equal to the free-space 
field E 0 if the slot is infinitely wide on both sides of 
M, which corresponds to a vector joining the two 
foci of the Cornu spiral. However, there is an infinite 
number of other finite openings of the slot which 
also will give the free-space field. Suppose, for 
instance, that h Q = h 0 ' in Figure 6 and that the slot 
width is gradually increased from zero. A glance 
at the Cornu spiral (Figure 3) shows that when 
v = 0.75 and v' = —0.75, the vector representing 
the diffraction field is approximately equal to the 
free-space field. This width represents, for a slot, 
the analogue of the first Fresnel zone for a circular 
opening. 

83 DIFFRACTION BY HILLS 

831 Introduction 

The formula for diffraction by a straightedge may 
be applied in radio practice to determine the diffrac¬ 
tion field behind a ridge. The ridge need not be 
perpendicular to the transmission path, but the 
condition given in equation (1) must be approx¬ 
imately fulfilled. The distance from transmitter 
and receiver to the ridge should be large compared 
to the height of the latter above the straight line 
TR; and that height should be large compared to 
the wavelength. 

Moreover, as pointed out in Section 8.1.3, the 
diffraction formula applies in principle only to the 
case where the effect of the currents induced on the 
surface of the ridge upon the field at the receiver 
can be neglected. This is the case (1) when the ridge 
has the shape of a steep and narrow knife-edge 
protruding from the surrounding countryside; or 
(2) when the surface of the ridge is rough (see Sec¬ 
tion 8.3.2). Experience shows that so long as the 
profile of the ridge is reasonably compact and its 
surface reasonably rough, the diffraction formula 
will give the magnitude of the field behind the ridge 
to within a few decibels. 



176 


DIFFRACTION BY TERRAIN 


If the ground near the transmitter or receiver is 
smooth, however, it becomes necessary to take 
ground reflection into account. This may be done 
by introducing an image transmitter and receiver. 
The field is then the sum of four components whose 
relative phase must be calculated (see Figure 11). 

Earth curvature will be neglected throughout the 
present section. 


8 3 2 Criterion for Roughness 

It is difficult to establish a quantitative criterion 
for the roughness of a surface. From the viewpoint 
of radiation theory, the effect of a rough surface is 
to scatter incident radiation diffusely in all directions 
with no preference for the direction of regular reflec¬ 
tion, whereas a smooth surface will reflect the inci¬ 
dent radiation according to Snell’s law. In radio 
work, the effect of diffuse reflection is to weaken the 
radiation scattered in the direction of the receiver 
so much that its intensity may be neglected com¬ 
pared to the direct ray. A moderately rough surface 
will give a coefficient of reflection intermediate 
between zero and unity. A surface will be optically 
smoother as the incident radiation approaches 
grazing, and even surfaces that are comparatively 
rough geometrically may then give partial reflection. 

A rule taken from optics and known there as 
Rayleigh’s criterion has been used successfully in 
radio practice. Assume that the roughness is pro¬ 
duced by numerous small elevations above a level 
surface and let H be the typical height of such an 
elevation. The difference in path between a ray 



Hence the critical value of H is given by 

47 rH\p TT jj \ 

-- = - , or H =-, 

X 4 16^ 


with \f/ in radians and 


„ 3.6X 


( 8 ) 


(9) 


with \f/ in degrees. The surface is considered smooth 
or rough according to whether H is smaller or larger 
than this value. 

Sometimes it is convenient to refer to the field 
pattern that would be present over a reflecting 
surface. This is done by introducing a new variable, 
the lobe number 


n-W 

X 


( 10 ) 


(hi transmitter height above the ground), where 
n = 1, 3, 5, etc., correspond to the angle of the first, 
second, etc., maxima in the lobe pattern and n = 0, 
2 , 4, etc., to the nulls of the lobe pattern. Introduc¬ 
ing n into equation (8), the criterion assumes the 
form 

hi 


H = ^ 


4 n 


(ii) 


Although the criterion is approximate and gives no 
more than an order of magnitude estimate, it is rather 
surprisingly well fulfilled in radio practice. Experi¬ 
ence has shown that when the differences in level 
which constitute roughness are of the order indicated 
by these equations, the reflection coefficient is 
reduced to a small fraction (about one-fifth) of the 
value calculated for an ideal surface. 

8 3 3 Diffraction by a Straight Ridge 

Assume that the ground intervening between the 
transmitter and receiver is everywhere rough, so 


Figure 7. Geometry for Rayleigh’s criterion for 
rough ground. 

reflected from the ground and a ray from the top 
of the elevation is 2 AB in Figure 7, which is equal 
to 2 H sin \J/ or 2H\p approximately for small angles i p. 
The difference in phase between the two rays is 
2H\1/(2tt/\). The criterion now requires that the 
surface be considered as rough when this phase 
difference exceeds 45 degrees, or 7r/4 radians. 



that all ground reflection may be neglected. For 
the sake of computation, {he ridge is replaced by a 
vertical screen of height h 0 above the line TR. The 






DIFFRACTION BY HILLS 


177- 


top of the ridge forms the diffracting edge (P in 
Figure 8). If the profile of the ridge is somewhat 
more complicated, the effective diffracting edge might 
be a purely mathematical line, as shown in the lower 
part of the figure. The height h 0 is conveniently 
determined from a profile of the transmission path 
obtained from a topographic map. If the heights 
hi, h 2 , and h of transmitter, receiver, and obstacle, 
respectively, above a given reference level such as 
sea level are given, we have 

h o= dih ;+f' ~ h , (i2) 

di d 2 

where the signs have been chosen so that h 0 is nega¬ 
tive when the receiver is in the shadow of the ridge 
and positive when it is in the illuminated region. 
Now, by equation (3), 

* = H'rfi + 4 ) = \l t° (ai +" 2) - (13) 

' X \di d / ' X 

In these equations, give the angles ai and a 2 the same 
sign as h 0 and give v the same sign that h 0 has in 
equation (12). 

The ratio of the field to the free-space field at the 
receiver is now given by | E/E 0 \ = z(v), defined by 
equation (7) and plotted in Figure 4. In Figure 9, 
this ratio is given in decibels as a function of the 
quantity x = —h 0 /^\d (all lengths in meters). 
The successive curves in Figure 9 correspond to 
different values of the ratio di/d 2 or d 2 /d\ (choose 
whichever one is the smaller). Only the field below 
the line of sight is shown. 

8 3 4 Field Near the Line of Sight 

The fact that just above the line of sight the field 
increases above its free-space value may sometimes 
be used to obtain a favorable site (Figure 10). The 
maximum value of the field is about 1.17 times the 
free-space value (Figure 4), equivalent to 1.36 db. 
On the other hand, there are advantages in avoiding 
a line TR that is too close to grazing the top of 
an intervening obstacle, as this will substantially 
reduce the signal. At the line of sight, the signal is 
6 db below free space. In order to get approximately 
the free-space value of the field, the crest of the 
obstacle should be sufficiently below the line TR 
so that v > 0.8 where v is given by equation (13), 
ho being the clearance between the line TR and the 
obstacle. In cases where the heights and distances 


are not quite certain, it is therefore preferable to- 
select a higher and definitely unobstructed site- 
rather than to try to utilize the small gain that might 
possibly be had from the diffraction field. 



0.15 0.2 0.3 0.4 0.6 0.8 1 2 3 4 6 


X—^-r- 

Ju 

Figure 9. Field in shadow behind a diffracting ridge. 



Figure 10. Diffraction field above the diffracting edge. 

8 3 5 Diffraction with Reflecting Ground 


When the ground near the transmitter or receiver 
is smooth and reflecting, the diffraction problem 
becomes very complicated. It can be solved by the 

















































178 


DIFFRACTION BY TERRAIN 


method of images on assuming that the radiation 
reflected on the transmitter side of the obstacle 
issues from an image transmitter and that the radia¬ 
tion reflected on the receiver side is incident upon 



Figure 11. Diffraction of both direct and reflected 
rays. 


an image receiver (Figure 11). The total field at the 
receiver may be written 

E = E\ — E 2 — Es El, 

where each term on the right-hand side is of the 
form of equation (6), Ei corresponding to the direct 
radiation, E 2 to the radiation from the image trans¬ 
mitter to the receiver, E 3 to the radiation from the 
transmitter to the image receiver, and FJ 4 to the 
radiation from one image to the other. These four 
terms differ in the value of v assigned to each of 
them; the effective height h 0 computed by equation 
(12) and the path lengths being different in each case. 

8 3 6 Example 

Assume that from a topographic map the profile 
shown in Figure 12 has been drawn. The horizontal 
scale is in kilometers and the vertical scale in meters 
above sea level. From this profile, combined with 
inspection of the terrain, it has been found that the 
ground is so rough that the reflected rays may be 

METERS ABOVE 
SEA LEVEL 



disregarded. The heights above sea level of trans¬ 
mitter, receiver, and obstacle, are respectively 
hi = 24 meters, h 2 = 33 meters, h — 69 meters. 
Since di = 9,000 meters, d 2 = 5,400 meters, d = 
14,400 meters, we find from equation (12) that 
-ho = — 39 meters. Assume a wavelength of 1 meter: 

x = ~ = 0.325 with ^ = 0.6. 

Vxd ck 


From Figure 9, the diffracted field is found to be 
14 db below the free-space field at the same distance. 

84 DIFFRACTION BY COASTS 
8,41 Introduction 

Diffraction occurring at coast lines is significant 
for coverage problems of coastal radars. It becomes 
particularly important when the sets are used for 
height-finding purposes where an accurate knowledge 
of the lobe angle and possible deformation of the 
lobes is required. 

The diffraction might be due either to the fact 
that the radar is sited on a cliff or to the sudden 
change in surface properties. Reflection from rough 
ground is diffuse, so that there is no interference 
between direct and reflected rays when the reflection 
point lies on this type of terrain, but interference 
does occur when the reflection point lies on the sea 
surface from which regular reflection is obtained. A 
situation commonly occurring is that of a search 
radar sited on rough terrain a fev r miles inland from 
the coast. Here coastal diffraction may result in an 
appreciable deformation (shortening or lengthening) 
of the lobes. 

More generally, diffraction occurs with level 
ground whenever there is a change, especial^ a 
sudden change, of ground properties along the trans¬ 
mission path. The formulas developed for coast¬ 
line diffraction may equally be applied to the case 
where rough ground suddenly changes into smooth, 
reflecting ground. Similarly, the effect of patches 
of smooth ground in rough surroundings, such as a 
lake in w r ooded country and, vice versa, rough 
patches in smooth terrain, may be treated by means 
of the Fresnel-Kirchhoff theory. Here, attention 
will be confined to the case of a straight boundary, 
applying the diffraction theory developed in Sec¬ 
tion 8.1. 

8 4 2 Level Site Near Coast 

Assume a transmitter sited on rough ground near 
a coast. If diffraction v r ere disregarded, the coverage 
pattern would appear as follows. When the reflec¬ 
tion point is on the land, the reflected ray is diffusely 
scattered and its field at the receiver is negligible. 
Again, if the reflection point falls on the sea, the 
reflected ray will be present and will interfere with 












DIFFRACTION BY COASTS 


179 


the direct ray with its full or nearly its full intensity. 
The ray leaving the transmitter at an angle \po 
(Figure 13), such that its reflected counterpart 
undergoes reflection right at the shore line, divides 
the coverage diagram into two parts. For angles of 



Figure 13. Diffraction by a coast line. 


elevation larger than \J/ = \f/ Q the field will be essen¬ 
tially the free-space field; for angles of elevation less 
than = \f/ 0 the familiar lobe pattern, for complete 
reflection, will appear with maxima equal to twice 
the free-space field. When diffraction by the coast 
line is taken into account, the discontinuity expressed 
by this rough picture is replaced by a smooth transi¬ 
tion of the field from one region to the other. 

The land surface may be considered as an opaque 
screen for the image transmitter from which the 
reflected rays seem to come (Figure 13). This prob¬ 
lem is somewhat different from the diffraction prob¬ 
lem treated previously since the trace of the screen 
in the vertical plane through T' and R is no longer 
perpendicular to the line T'R as it was, for instance, 
in Figure 2 , upper part. In the present case, the 
effective height ho of the diffracting edge for any 
given ray is the perpendicular projection from the 
coast line upon this ray, as shown in Figure 13. 
The slant distance of the coast from the trans¬ 
mitter is d\. Assuming that the receiver (target) is 
far distant, a condition usually fulfilled in radar 
practice, d 2 >> di and the angle \p between the 
direct ray and the horizontal will be equal to the 
angle between the image ray and the horizontal. 
Then approximately, since the angles are small, 

h 0 = di«i = di(\J/ 0 — i/0, (14) 

where di and ai have the significance given them in 
Section 8.1.4. Here the signs have again been 
chosen so that h 0 is negative when the receiver 


(target) is in the shadow of the screen with regard 
to the image transmitter. 

The distance from the transmitter to the diffract¬ 
ing coast depends on the azimuth (Figure 13). 
Therefore, with the designations of the figure, 

* - — . (15) 

COS 7 


8 4 3 Equation for Field Strength 


The expression for the diffracted field of the 
image transmitter is given by the straightedge 
formula, equation ( 6 ), with v given by equation (3). 
Since 1 /d 2 is assumed negligibly small compared to 
1/dij we find, on using equation (14), 

V = (*0-*)>|y 1 - ( 16 ) 


This may be further simplified by introducing (as in 
Section 8.3.2) a new variable, the lobe number 


4fti» 

X 


(17) 


(hi = transmitter height). This quantity is equal to 
1, 3 , 5 • • • at the interference maxima and equal to 
0, 2, 4, • • • at the interference minima but is here 
taken as a continuous variable, defined for any value 
of \p. In particular for \f/ = \po we put n = n 0 . Since 
\po = hi/di t we have by equation (15) 


4/q^o 4/u 2 cos 7 

X Xdo 


(18) 


Equation (16) may now be written 


n 0 — n 
V2 no 


(19) 


The diffraction formula will again be written, in the 
form of equation (7), as 


E_ 

Eo 


= ze 


where z and f are the functions of v shown in Figures 
4 and 5. 

The total field obtained by the interference of the 
direct and reflected ray is 


E = Eo(l - ze- jnn - j *) y (20) 

where the negative sign in front of the second term 
in parentheses accounts for the 180-degree phase 
shift at reflection, and the phase lag irn corresponds- 










180 


DIFFRACTION BY TERRAIN 


to the path difference between the reflected and 
direct rays. 

The absolute value of the field is 


E_ 

E 0 


V (1 — z) 2 + 4 z sin 3 ^(irn + £). 


( 21 ) 


Figure 12 in Chapter 5 may be used for the numer¬ 
ical evaluation of this equation. 

The formula can readily be generalized to the 
case where the reflected ray is weakened by ( 1 ) a 
reflection coefficient, R, different from unity, and 
( 2 ) the effect of the earth’s curvature expressed by 
the divergence factor, D, (Chapter 5). If, moreover, 
the phase lag at reflection is not 7 r but t + </>', the 
equation becomes 

= V(1 — zRD) 2 + 4zRD sin 2 ( 7 m + </>'+ f). 

( 22 ) 


E 

E 0 


84 4 Example 

Assume the following conditions. A radar set of 
200 me (X = 1.5 meters) is sited at a height hi = 15.3 
meters (about 50 feet) and at a distance to a straight 
shore line of d 0 = 195 meters (about 0.12 mile). 
The ground between the radar and the seashore 
is level but can be considered as rough for prac¬ 
tically any angle of elevation, on applying the 
criterion of Section 8.3.2. The coverage diagram 
will first be determined in the azimuth perpendicular 
to the coast line, where d\ = d 0 , or cos 7 = 1 . 
Then by equation (18), n 0 = 3.20. With this value 
of n 0 the variable v is determined by equation (19). 
We shall confine ourselves to integral values of n, 
that is, to those angles which, in the presence of 
simple reflecting ground, correspond to lobe minima 
and maxima. Having obtained v, one then deter¬ 
mines z and £ from Figures 4 and 5. The field in 
terms of the free-space field is then obtained from 
equation ( 21 ), either by direct computation or by 
means of Figure 12 in Chapter 5. The numerical 
data for the first five lobes are summarized in 
Table 1. The last column of this table contains the 
values oi E/E 0 which would be obtained if the magni¬ 
tude of the reflection coefficient were assumed to be 
zero over land and unity over the sea and if diffrac¬ 
tions were neglected. 

The same calculations are carried out for an 
azimuth inclined by an angle 7 = 45° with respect 
to the coast line. Then, from equation (18), n 0 = 4.5. 
The results are given in Table 2 . 


It is seen from these data that the lobes near the 
critical ray (ray whose reflection point is at the coast 
line) undergo very considerable deformation. The 

Table 1. ( 7 = 0 °). 


n 

V 

z 

r 

(degrees) 

\E/Eq\ with 
diffraction 

\E/E 0 \ without 
diffraction 

0 

1.27 

1.17 

0 

0.17 

0 

1 

0.87 

1.05 

- 12 

2.05 

2 

2 

0.48 

0.80 

- 15 

0.75 

0 

3 

0.08 

0.54 

- 4 

1.54 

2 

4 

-0.32 

0.36 

24 

0.69 

1 

5 

-0.71 

0.26 

11 

1.26 

1 

6 

- 1.11 

0.19 

145 

1.16 

1 

7 

- 1.51 

0.14 

242 

0.94 

1 

8 

- 1.90 

0.12 

5 

0.88 

1 

9 

-2.30 

0.10 

158 

0.92 

1 

10 

-2.70 

0.08 

339 

0.94 

1 




Table 2. 

(7 = 45°). 






\E/Eo\ with 

\E/Eo\ without 

n 

V 

z 

r 

(degrees) 

diffraction 

diffraction 

0 

1.50 

1.07 

6 

0.10 

0 

1 

1.17 

1.17 

- 3 

2.17 

2 

2 

0.83 

1.03 

- 13 

0.21 

0 

3 

0.50 

0.82 

- 15 

1.80 

2 

4 

0.17 

0.59 

- 8 

0.42 

0 

5 

-0.17 

0.42 

11 

1.42 

1 

6 

-0.50 

0.31 

42 

0.80 

1 

7 

-0.83 

0.23 

90 

0.91 

1 

8 

- 1.17 

0.18 

157 

0.88 

1 

9 

- 1.50 

0.14 

240 

0.99 

1 

10 

- 1.83 

0.12 

338 

1.12 

1 


coverage pattern corresponding to Table 1 is shown 
graphically in Figure 14. 



Figure 14. Coverage diagram (relative field strength). 
(Heights exaggerated 3.5 to 1.) 


In the problem considered here, the angles of 
elevation are comparatively large (for n — 1 , 
\f/ = 1 ° 24'). If the effects of diffraction occur at 






















DIFFRACTION BY COASTS 


181 


lower angles, the divergence factor D must be taken 
into account (see Chapter 5). This is done by com¬ 
puting D for the angles desired and replacing z by zD 
in equation (21). 

84 5 Cliff Site 

If the radar is sited on a cliff and if the land inter¬ 
vening between the radar and the reflecting plane 
(ocean) is rough, the equations of Section 8.4.3 
apply. We shall now consider the case where the 
radar is sited at some distance from the cliff edge 
and where the ground between the radar site and the 
cliff edge is reflecting. There are then two reflecting 
planes, the lower of which might be the ocean, or 



Figure 15. Diffraction from a cliff site. 


might be a reflecting land surface. In Figure 15, 
this surface has been designated as ocean. The upper 
plane is at a height H above the lower plane and the 
transmitter at a height hi above the upper plane. 
Assume that the azimuth chosen is perpendicular 
to the direction of the cliff edge; the distance of the 


radar to the cliff edge is d 0 . For any other azimuth 
(angle y in Figure 13), replace do by d 0 /cos y in the 
following equations. 

Two images are shown in Figure 15 and two 
fictitious opaque screens, one corresponding to each 
image. The corresponding variables are distinguished 
by single and double primes. The lobe numbers are 
given by 

n" = 

X 

n , = m + W = n"{H + hi) 

X hi 

The critical angle, \f/ 0 = hi/d 0) is the same for both 
image transmitters. Thus 


, _ 4(ff + h)hi _ no"(H + hi) 

no — - —-- 

Xdo hi 

Further 

v , _ n 0 ' — n' 

vw 

V2no" 1 + H/h 

Again, the field is given by 

E = E 0 (l - z'e~ jnn '~ K ' - z"e~ jnn "~ K "). (23) 

The expression for the field strength [equation (23)] 
is in a form where all the quantities involved may be 
evaluated for any given height of transmitter, height 
of the cliff, and any wavelength by using graphs and 
tables given in earlier paragraphs. 
















Chapter 9 
TARGETS 


SCATTERING PARAMETERS 
Radar Cross Section 


I n determining the coverage to be expected of 
radar systems, it is important to know what 
fraction of the power incident upon a target will be 
returned to the receiver. A parameter involving the 
dimensions and orientation of the target, and usually 
also the wavelength, and which measures the propor¬ 
tion of power returned, is called a scattering para¬ 
meter. 

The most generally used of these parameters is the 
radar cross section introduced in Section 2.4.1. It 
is denoted by a and is defined by 


a 


= 4:ird 2 


W r 

W<’ 


( 1 ) 


where W r is the scattered power per unit area at the 
receiver and Wi is the incident power per unit area at 
the target. In terms of a, the radar gain is 



This equation may also be written in the form 


where 




is the gain factor introduced earlier (see Section 5.1), 
A 0 is the free-space gain factor and A p is the path- 
gain factor. 


912 Target Gain 

Another scattering parameter is G R , the target 
gain, discussed in Section 2.4.2. It is the gain of the 
target in the direction of the receiver relative to a 
shorted (dummy) doublet antenna. The target gain 
is connected with a b} r the relation 

(3) 

O X 


The corresponding radar gain is 

^ = 4 G 1 G 2 G b *A\ (4) 

Pi 

The factor 4 is due to the calculation of Gr relative to 
a shorted doublet rather than to a matched load 
doublet. If the calculation of Gr were made relative 
to the matched load doublet, the factor 4 would be 
replaced by 1. 


9.1.3 


Echo Constant 


The echo constant, denoted by K, is defined by 


K = ^( d V 


w ( \x, 

and is related to a by 

K = —°—. 
4tt\ 2 

The corresponding powder ratio is 

P 2 


Pi 


= KGiGi 


(?)- 


(5) 


( 6 ) 


(7) 


Except for the factor /4tt, K is just a measured 
in square wavelengths. 


914 Equivalent Plate Area 

A plate of area S placed normal to the direction of 
propagation has a radar cross section given by 

<r = 4x-, (8) 

X" 

provided the linear dimensions of the plate are large 
compared with X. Any target may be supposed to 
scatter (in the direction of the radar) an amount of 
energy equal to the amount that a plate of area S 
would scatter in this direction. This area S is called 
the equivalent plate area of the target. The corre¬ 
sponding radar gain is 



482 



RADAR CROSS SECTION OF SIMPLE FORMS 


183 


915 Scattering Coefficient 

or Characteristic Length 

This parameter has also been called the radar 
length of the target. The definition is 

L = d^, (10) 

where E r = field strength at the receiver, 

Ei = field strength incident on target. 

It is evident that 

L 2 =-f- (11) 

4 7T 

connects L with radar cross section. The radar gain 
becomes 

92 RADAR CROSS SECTION 

OF SIMPLE FORMS 

921 Spheres 

The radar cross section of any large curved con¬ 
ducting surface having principal radii of curvature 
Pi and p 2 at the reflection point is given by 

<j = xpip 2 . (13) 

This formula applies if the surface is sufficiently 
large and sufficiently curved to contain many 
Fresnel zones. For a sphere of radius a, where 
a >> X, 

a = 7r a 2 . (14) 

Thus, in the case of a large conducting sphere, the 
radar cross section is equal to the geometrical cross 
section and is independent of wavelength. 

The result for small spheres (a < < X) is 

<7 = 1447T 5 — . (15) 

X 4 

There is no simple formula for the radar cross 
section in the region a ~ X. 


922 Cylinders 

The radar cross section of a cylinder whose length 
is large compared with the wavelength is 

2t aL 2 


(16) 


where 

a = radius, 

L = length (L > >X). 

This formula assumes that the direction of inci¬ 
dence is normal to the cylindrical surface. If the 
cylinder is tilted so that there is a small angle 6 
between the normal to the cylinder and the direction 
of incidence, the result is 


2tt aL 2 

a =- 

X 

This result holds for small angles of tilt 6 such that 
sin 0 ^ 0 . 


sin- 


2 7 tL9 


2tL6 


(17) 


Plates 


A flat plate of area S with all dimensions large 
compared with X and oriented so that the normal 
to the plate is in the direction of incidence, has a 
radar cross section given by 

£ 2 

(7 = 4tt- , (18) 

X 2 


regardless of shape. 

For a circular plate (a disk) of radius a, whose 
normal is at an angle 6 with the direction of incidence, 


1 [cot d • Ji ^sin o'j J > 


where J i is the first-order Bessel function, 
maximum value is at 6 = 0 , where 


(19) 

The 



( 20 ) 


This agrees with equation (18), since at normal 
incidence S = t a 2 . 

The peculiar feature of equation (19) is that the 
maximum at 6 = 0 is very sharp. For example, if 
X/a = 1/10, <7 is only 1/10 of its maximum value 
when 6 = 1.25°. 

The average value of cr over all orientations is 



This result is independent of wavelength and sug¬ 
gests that a large number of flat plates oriented at 
random will have a cross section independent of X, 
or that a few surfaces of rapidly changing orienta¬ 
tion may have this property. 








184 


TARGETS 


210 ° 200 ° 190 * - 180 ° 170 ° 160 ° 150 ° 



Figure 1. Aspect diagrams of B-17E, 5 degrees above horizon. 






























AIRCRAFT 


185 


The results for a rectangular plate are practically 
the same as for a disk. If the dimensions of the 
plate are b and c, ira 2 is replaced by be in equation 
(20) and equation (21); equation (19) is replaced by 


4t rb 2 c 2 


X 2 


cos 0- 


(2irb . . A 

I —sin 0 • cos 0 I 

~2irb . . T~ 

—— sin 0 • cos 0 
X 



sin 0 • sin 0 


) 


2tc 

IT 


sin 0 • cos 0 


( 22 ) 


where the sides 6 and c are parallel to the x and y 
axes, and sin 0 cos 0, sin 0 sin 0, and cos 0 are the 
direction cosines of the direction of incidence rela¬ 
tive to the plate normal. 

These results hold when the linear dimensions 
of the target are large compared with the wave¬ 
length. If the linear dimensions are small compared 
with the wavelength, a plate of area S, oriented so 
that the normal to the plate is in the direction of 
incidence, has a radar cross section given by 


32 7r 2 S 3 
~3~ X 4 ‘ 


(23) 


Corner Reflectors 


The corner formed by three mutually perpen¬ 
dicular conducting planes forms w r hat is called a 
corner reflector. The faces may be triangular or 
square or have other shapes, depending on how the 
planes are bounded. A line drawn to the corner 
making equal angles with the three edges is called the 
axis of symmetry. 

Reflection from a corner reflector may be analyzed 
by the methods of geometrical optics, provided the 
linear dimensions of the reflector are large compared 
with the wavelength. A ray which is reflected from 
all three surfaces is said to be triply reflected. 
Triply reflected rays always return to the radar and 
make the only large contribution to the radar cross 
section. The radar cross section of a corner reflector 
is 


cr 


4thS 2 

“x 2 "' 


(24) 


where S is the cross section of the triply reflected 
beam. S is a function of the shape of the faces of the 
corner reflector and of the angle of incidence of the 
radiation. 


For a triangular corner reflector, a is given approx¬ 
imately by 

4 7 rTA 

(7 = (1 - 0.00076 0 2 ), (25) 

3X 2 

where L = length of edge of reflector, 

0 = angle between direction of incidence and 
the axis of symmetry in degrees (0 < 26°). 
As a function of 0, a has a broad, flat maximum. 
Consequently, the return to the radar receiver from 
such a target is not sensitive to the precise orienta¬ 
tion of the axis of symmetry. 


93 AIRCRAFT 

9,3,1 Variation with Aspect 

Diagrams showing the dependence of a on orienta¬ 
tion indicate very large and irregular fluctuations. 
Radar cross section a can change from values of 
nearly 1,000 square meters to a few square meters as 
a result of a change of aspect of a few degrees. These 
instantaneous values of the radar cross section are 
dependent on wavelength, polarization, details of 
plane design, areas of specular reflection, propeller 
rotation, etc. Reflection patterns have been meas¬ 
ured for a few simplified models by laboratory means 
(see Figure 1 as an example). It would be difficult 
to calculate instantaneous values of <r by theoretical 
methods. 

In practice, however, an airplane is in motion and 
is affected by air currents. These factors cause the 
airplane, in a short interval of time, to present many 
widely different instantaneous values of a to the 
radar, so that the signal actually seen on the scope 
by the observer is in effect a time average, where the 
most violent fluctuations of instantaneous values 
of <t have been smoothed out. 

9 - 3,2 Measurement of cr 

The radar equation for free space, equation (45), 
in Chapter 2, may be used for the computation of 
average values of <t from observed instantaneous 
values, provided conditions are such that ground 
reflections are unimportant. The received power Pz 
is determined by matching the signal from the plane 
with the measured signal from a signal generator. 

The procedure followed in w'ork at the Radiation 
Laboratory is to measure the maximum value of P<i 








186 


TARGETS 


for each of a series of 3-second intervals. A plot is 
made of P 2 against range d on log log coordinates. 
As might have been anticipated from equation (45), 
in Chapter 2, it is found that a line with a constant 
slope of —4 passes through the average of the 3- 
second interval maximum points, although the 
individual points fluctuate widely. The value of a 
corresponding to this line is calculated. 

The resulting value of a still cannot be called an 
average value because the maximum value of a 
has been used for each point. Consequently these 
values of <7, substituted into equation (45) in Chap¬ 
ter 2, cannot be expected to give the average value 


of P 2 , or to give observed maximum ranges. How¬ 
ever, it is found that if the values of a thus computed 
are reduced 40 per cent, they give correct results. 

These empirical cross sections are relatively inde¬ 
pendent of wavelength. This result may be inter¬ 
preted to mean that a plane in motion behaves more 
or less like a collection of specularly reflecting sur¬ 
faces oriented at random, as equation (21) indicates. 

Attempts have been made to develop formulas 
giving operational cross sections as a function of 
some large feature of plane design, such as wing 
span or length of fuselage, but these attempts have 
not been successful. 



Chapter 10 
SITING 


101 GENERAL 

1011 Introduction 

S iting refers to the selection and utilization of 
local terrain features which affect propagation 
and the performance of equipment. From a pre¬ 
liminary analysis, the general location, type of 
equipment, and height may be determined. The 
specific sites available may, however, profoundly 
alter performance in several ways. Careful analysis 
and tests may then be necessary to determine the 
best use of the facilities at hand and for an under¬ 
standing of the limitations due to the terrain. 

10,1,2 Siting Requirements 

With communication equipment, the siting prob¬ 
lems are principally concerned with visibility and, 
in wooded areas, absorption by vegetation. When 
siting direction-finding equipment, it is important 
to realize that reflections from mountains or other 
irregularities may cause serious angular errors which 
should be avoided by proper choice of the location. 
Both direction-finding and radar equipment require 
orientation. 

Radar siting requirements are rather different and 
depend on whether ground reflection is of importance 
or not. The siting of radars operating mainly on 
the direct ray is relatively easy and is principally 
concerned with permanent echoes and visibility. 
The most exacting site requirements are presented 
by the VHF early warning and height-finding radars, 
which to a large extent depend on ground reflection 
for successful operation. The siting problem then 
requires the consideration of terrain effects such as 
limited reflection areas, cliff edges, obstacles, etc., 
which involve diffraction problems of considerable 
complexity. Recommendations for specific sets are 
given in instruction manuals furnished with the 
equipment. 

10 2 TOPOGRAPHY OF SITING 

10,3,1 Maps 

Radar and direction-finding systems, which may 
cover a large area and involve many services, use a 


grid for plotting purposes. The grid location, height, 
and orientation of each station must be known with 
reasonable accuracy. Topographic maps of a scale 
of one or two miles to the inch and contour intervals 
of not more than 100 feet, preferably 20 feet, should 
be secured. These may be supplemented by aerial 
photographs and surveys. 


10 2 2 Profiles 

In a complicated terrain, it is usually necessary to 
have profiles on several azimuths to determine the 
effective height above the reflecting surface. The 
accuracy required decreases with the distance from 
the transmitter. In most cases sufficient detail is not 
available on maps, so that a personal inspection of 
the terrain should be made to become familiar with 
the nature of the soil and degree of roughness. 
Special attention should be given to ridges, flat 
areas, bodies of water, distance to the shore, hills 
to the rear, obstacles in the operating area and 
at the boundaries. 


10 2 3 Orientation 

Where long distances and directive beams are 
involved, fairly accurate orientation of the order of 
one-half degree is required. Care must be taken when 
using compasses because of local attractions or 
inadequate information on declinations. Observa¬ 
tions on Polaris give the greatest precision but this 
star is not always visible and it is often inconvenient 
to use a transit at night. Caution must be used in 
aligning on permanent echoes, as they may be diffi¬ 
cult to identify. In general, several methods should 
be used to obtain independent checks. 

Solar azimuths, correct to the nearest quarter of a 
degree, may be determined from the date time to the 
nearest minute, and the latitude and longitude to 
the nearest degree. Two methods will be given for 
obtaining the azimuth of the sun: (1) by calculation, 
(2) from tables. A third method gives true south 
only. 


187 


188 


SITING 


The azimuth of the sun may be calculated from 
the formula, 

sin (HA) 


tan 0 = — 
where (3 


( 1 ) 


HA = 


</> = 
8 = 


cos <f> tan 8 — sin</> cos (HA) ’ 
bearing of the sun. The bearing is east or 
west of south when <f> — 8 is positive. 
The bearing is east or west of north when 
(f) — 8 is negative. The bearing is east 
in the morning (0 will be negative) and 
west in the afternoon (0 will be positive), 
hour angle of the sun. During the morn¬ 
ing hours when the hour angle is greater 
than 12 hours, its value should be sub¬ 
tracted from 24 hours for use in the 
formula. 

latitude of the place of observation, 
declination of the sun at the time of 
observation. The signs of <p and 5 are 
important and each is positive when 
north of the equator and negative when 
south. 

The hour angle HA is the local apparent time 
(LAT) minus 12 hours. To convert the observed 
time into LAT, the civil time at Greenwich (GCT) 
must be found and combined with the equation of 
time to correct for the apparent irregular motion of 
the sun. This gives Greenwich apparent time 
GAT, which is converted to LAT, by allowing for 
the longitude. The equation of time and the decli¬ 
nation of the sun are plotted for 1945 in Figure 1. 
The annual change is small and these curves may 
be used for most orientations without regard to the 
year. Standard time meridians are given every 
15 degrees east or w T est of Greenwich, each zone 
corresponding to one hour. Care should be used to 
take daylight saving or other changes from standard 
into account correctly. 

The calculations may be illustrated from the 
following data: date, 16 March; time, 1345 hours 
PWT; latitude, 40° north; longitude, 118° west. 
The HA is computed first. 

Observed time (PWT) 13 hr 45 min 

Zone difference + 7 hr 


Greenwich civil time 

20 hr 

45 min 

Equation of time (Figure 1) 


- 9 min 

Greenwich apparent time 

20 hr 

36 min 

Longitude difference (for 118° W) 

- 7 hr 

52 min 

Local apparent time (LAT) 

12 hr 

44 min 

LAT —12 hours = H A 

- 12 hr 


Hour angle of sun 

+ 0 hr 

44 min 

HA in arc 

+ 11° 


Latitude 4> 

+ 40° 


Declination of sun 8 (Figure 1) 

- 2° 



Substituting in equation (1), 


cos 40° * tan ( — 2°) — sin 40° • cos 11° 

0 = 16° 10' 

Since <j> — 8 is positive, 0 is the bearing from the 
south. The bearing is west of south, since HA is 
positive (p.m.). The azimuth of the sun is 
180° + 16° 10' = 196° 10'. a 

The equal altitude method is less convenient but 
requires no calculation. This method consists in 
measuring the horizontal angles between the sun 
and a mark taken when the sun is at the same alti¬ 
tude on both sides of the meridian of the observer. 
The bisector of the horizontal angle between the 
two equal altitude positions of the sun during the 
observations is very close to true south, and the 
azimuth of the mark may be determined. 


10 3 GEOMETRICAL LIMITS OF VISIBILITY 


Horizon Formula 


It is assumed throughout that the earth radius is 
ka (see Section 4.1). Whenever numerical examples 
are given, the standard value, k = 4/3, is used. 
The alternate method of accounting for refraction 
given in Section 4.1.5 may also be used in connection 
with the following equations if k 7^ 4/3. 

When a horizontal ray, tangential to the earth, is 
drawn, the earth slopes aw r ay (Figure 2) at the rate of 


h = 


2ka 


( 2 ) 


Hence the horizon distance dr for a transmitter at a 
height h above level ground is equal to 

d T = V 2kah . (3) 

Numerically, when all the lengths are in meters 
_ 4 

d T = 4,120 Vfc for k = - . (4) 

With h in feet and dr in statute miles, by a curious 
numerical coincidence, 

dr — V 2/1 for& = ~* 0>) 


a This result could have been obtained directly from 
Azimuths of the Sun, H071, U.S. Naval Department, Hydro- 
graphic Office. The equation of time may be obtained from a 
current copy of The American Nautical Almanac, U.S. Naval 
Observatory, Washington, D.C. 










GEOMETRICAL LIMITS OF VISIBILITY 


189 


When both terminals of a path are elevated above 10 3 2 Height of Obstacle 

the ground (Figure 3), the horizon distance is 

_ _ As a first case, consider a smooth earth and two 

dx = V2 ka (V/h -b V/ 12 ), (6) terminals at the ground. The earth itself forms an 



1 11^21 3M 0 ^20„2 12 22,1 II 21. ,1 II 21 31. JO 20 30.10 20 30.9 19 29 8 18 28, 8 18 28,7 17 27 7 17 27, 

JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV OEC 

SUN DATA FROM NAUTICAL ALMANAC 1945 

Figure 1 . Calculation of solar azimuth. 



Figure 2. Geometry for horizon distance for zero 
height transmitter. 


where again V2/ca = 4,120 in the metric system. 
If d is in statute miles and h in feet, 

d L = yl2th + V2 h 2 for k = (7) 

The relation between hi, h 2 and d L is graphical!}^ 
presented in Chapter 5 in the form of a nomogram, 
Figure 2. 


T 



Figure 3. Geometry for horizon distance with ele¬ 
vated transmitter. 


obstacle which reaches its maximum height h m in the 
middle of the path (Figure 4). By equation (2) 



A point P on the ground at distances d' and d" from 
the two terminals (see Figure 4) has an elevation 
























































































100 


SITING 


above the straight line connecting the terminals 
given by 

k = ' 2 l - ^ 2 ’ (9) 

2 ka 2 ka 


while the second term again becomes d'd" /2 for 
k = 4/3. 

Equation (13) is used to decide whether and by 
how much an obstacle such as a hill will obstruct a 
given transmission path. 


or, after a simple reduction, 

h = — = 5.9 • 10 s d'd", (10) 

2 ka 


2 

1 

P 

ih^\ 



L_ rT _ 


- —— U 

*- - d 

' 0 J 


Figure 4. Height of earth as an obstacle. 


where h, d', and d" are given in meters. If h is in 
feet and d' in statute miles, 
d'd" 

h = ^~ for k = 4/3. (11) 

Secondly, assume that the terminals are elevated 
(Figure 5). The elevation of the straight line con¬ 
necting the terminals, for a flat earth, is equal to 
d% - d'% 


h = 


d' 


( 12 ) 


where d', d" are again the distances to the terminals, 
and h h h 2 are the corresponding elevations. 

In order to account for the effect of the earth’s 
curvature, Figure 5 may be considered as a plane 
earth diagram on which a ray will appear curved, the 
deviation from a straight line being downward and 
given by equation (10). This is indicated by the 
dashed line in Figure 5. 



U-d' — *** d" —*1 

Figure 5. Height for elevated terminals. 


Hence, the total height above the theoretical 
ground is 


d'h 2 - d"hi d'd" 

d' — d" 2ka 


(13) 


When the heights are expressed in feet and the dis¬ 
tances in miles, the first term remains unaltered, 


10 3 3 Extended Obstacle 

When the obstacle is of appreciable horizontal 
extension, it may not possess a single peak to wdiieh 
equation (13) can be applied without ambiguity. 
The case of twin mountains is shown in Figure 6 for 
straight rays (earth’s radius ka). 



The optical peak P of the obstacle for radio or radar 
transmission is the point from which both terminals 
are just visible. For a given profile, the limiting rays 
to the terminals may be found by trial and error by 
applying equation (13) to those points of the profile 
which are most likely to represent limiting elevations. 
In the theory of diffraction given in Chapter 8, 
P marks the position of the equivalent diffracting 
edge. 


10 3 4 Degree of Shielding 

As a measure of the degree of shielding, the angle 
between the two limiting rays drawn from the 
terminals to the (actual or equivalent) peak of the 
obstacle of height h p may be used. 

Since all angles considered are small, the sine or 
tangent of the angle may be replaced by the angle 
in radians. Consider first the ray going from the 
first terminal to P (Figure 7). The angle of the ray 
with the horizontal at the terminal is 


ai = 



d/_ 
2 ka 


(14) 


and its angle with the horizontal at P is 

. . d' h p — hi d' 

ft = “ 1 + fa =^r~ + Zka 


(15) 


The angle of the ray going from the second terminal 
to P is determined correspondingly. 






























PERMANENT ECHOES 


191 


The angle between the two rays is then equal to 


Pi + P 2 


/ _i_ I K_ _ ^ 
\2 hand'd" d' 



where d = d' + d", and (ha)' 1 = 1.18 • 10 ' (meter)" 1 . 
When d is measured in miles and h in feet, equation 
(16) becomes 


ft + ft = 1.89 • lO^A + 1 - - - - 1. (17) 

L d’d" d’ d" J 



Figure 7. Shielding between transmitter and receiver. 


4. Strong permanent echoes from mountains to 
the rear may be caused by back radiation from the 
antenna. The low intensity of the back radiation 
may be compensated by the size of the mountains. 
Such echoes are especially harmful as they obscure 
the operating sector. 

5. Objects appear wider because of the antenna 
beam width and of greater extent in range as a result 
of the pulse width. 

6. Diffraction over intervening ridges may be 
sufficient to nullify their screening action so that 
objects behind a ridge are visible. 

7. The use of a permanent echo as a standard 
target may be very misleading. A decrease in per¬ 
formance that seriously affects echoes from small 
targets may not have any noticeable effect on the 
response from large targets. An echo used for a 
standard target should be weak and near by. 


104 PERMANENT ECHOES 

1041 Introduction 

Permanent echoes are caused by reflections from 
terrain features such as mountains or even smooth 
surfaces near the antenna (ground clutter). With 
radars, the indicator is obscured by the strong- 
echoes from hills and the minimum detection range is 
increased by ground clutter. With direction finders, 
erroneous indications are caused by the spurious 
reflections. Permanent echoes are among the princi¬ 
pal problems involved in siting, as many otherwise 
excellent sites are rendered worthless by excessive 
fixed echoes. Several methods are available for 
determining the suitability of sites in this regard 
without actual field tests. 

A number of factors combine to make permanent 
echoes more troublesome than might be expected. 

1. Hills and land surfaces are so much greater in 
extent than the target which the equipment is de¬ 
signed to detect that strong echoes may be obtained 
from distances where an ordinary target would give 
an echo far below normal detection levels. 

2. The low elevation of the land surfaces places 
them in regions most subject to nonstandard 
propagation effects where extreme ranges and large 
responses are frequently obtained. 

3. Side lobes of the horizontal pattern of the an¬ 
tenna cause permanent echoes to appear at several 
other azimuths in addition to that of the main lobe. 


10.4.2 Permanent Echo Diagrams 

The permanent echoes associated with a radar 
station may be plotted on a polar chart and their 
extent, location, and strength represented. Such 
diagrams should be prepared for each unit of a radar 
system, using a standard procedure for taking and 
presenting the data. 

Permanent echo data should be taken under 
average conditions with the gain set at some stand¬ 
ard level. At intervals of azimuth such as 5 de grees,. 
the ranges of the permanent echoes are recorded. 
These data are then plotted on a polar chart and the 
points are connected to indicate obscured areas. 
The skill and judgment of the operator are important 
factors. In most cases the amplitudes of the echoes 
are so far above that of ordinary target echoes that 
the actual amplitudes need not be noted. 

In Figure 8 is shown an observed permanent echo 
diagram for a VHF radar. This was selected for 
purposes of illustration rather than as an example 
of a good site. The mountains to the north are un¬ 
shielded and cause extensive echoes. The large echo 
at 200 degrees is due to a mountainous island 260 
miles away and appears only during times when 
propagation is nonstandard. 

Care must be exercised in identhying the cause of 
an echo. Antenna side lobes cause spurious echoes 
and distant echoes may come in on the second or 
third sweep on the scope after the main pulse. These 
latter echoes may be checked by changing the pulse 
repetition rate and observing the shift of the echo. 






192 


SITING 


Permanent echo diagrams are useful for: 

1. Indicating blind areas in a station’s coverage. 

2. Assigning the operating area of a station. 

3. Checking the range and azimuth accuracy. 

4. Checking the performance. 

5. Estimating nonstandard propagation. 

G. Planning test flights. 


hills are screened by a local obstruction. This local 
echo at, say, three miles, is combined with the main 
pulse or ground return and the distant echo is 
weakened or eliminated entirely. 

Shielding causes a loss of coverage, which in 
operating regions may be more serious than the 
permanent echoes. Rear areas which are not scanned 



Figure 8. A typical permanent echo diagram for a VHF radar. 


10 43 Shielding 

The principal device in the field for the control of 
permanent echoes is shielding. This means that the 
antenna must be sited in such a way that distant 


should be well shielded so that back and side echoes 
do not interfere with targets in important tactical 
regions. Operation over such shielded sectors would 
be limited to high targets. 










































































PERMANENT ECHOES 


193 


10.4.4 p re di c ti 0n of Permanent Echoes 

Permanent echoes may be determined by several 
methods: (1) tests with the radar at the site; (2) radar 
planning device [RPD]; (3) supersonic method; and 
(4) profile method. 

The feasibility of moving the radar to the site to 
determine the permanent echoes is dependent on the 
portability, accessibility, etc. Echoes obtained with 
one type of equipment may be very different from 
those of another type of radar with a different an¬ 
tenna directivity, frequency, and range. 

The RPD technique requires construction of a relief 
model of the terrain considered. A small light source 
is used to simulate the radar transmitter and the 
echoes are plotted as a result of a study of the areas 
illuminated. This method is useful for short ranges 
and microwaves where the diffraction and side and 
back lobe radiation are small. Construction of a 
fairly difficult relief model may take a crew of 
specially trained men several days to a week, as the 
model should be accurate. Once completed, all 
possible sites or aspects from a plane or ship may be 
readily examined. Models of enemy areas may be 
used to predict the coverage of possible enemy sites 
and evasive action may be planned. The RPD is 
well suited for training and briefing of air personnel. 



Kits are provided containing the light source, sup¬ 
ports, etc. Photographic and darkroom facilities 
are also required. 

The supersonic method uses a relief model under 
water. Supersonic gear is used to send out pulses 
which are reflected like radar pulses and an echo is 
picked up and presented on a plan position indi¬ 
cator [PPI] scope. Photos may be taken of the 
scope or it may be used directly to train operators 
and for briefing. Considerable equipment is re¬ 
quired, but the construction of the models is com¬ 
paratively simple. 

The profile method involves a study of topo¬ 


graphical maps and plotting of the echoes according 
to their visibility and the amount of diffraction. A 
fairly difficult site may be handled in perhaps eight 
man-hours. This method is adapted to long-range, 
low-frequency radars where diffraction and side and 
back lobe radiation are important. On microwave 
equipment, prediction of permanent echoes is sim¬ 
pler and the profile method may be worked out in 
a few hours. 

10.4.5 p re di c ti on by the Profile Method 

The discussion here refers chiefly to VHF (1 to 10 m) 
radars in a mountainous terrain, but the methods 
have general application. The principal requirements 
are topographic maps of the surrounding area with 
a scale of one or two miles to the inch and a contour 
interval of 20 feet, although intervals up to 100 feet 
may be used. Regional aeronautical maps with a 
scale of about 1 inch to 16 miles and 1,000-ft 
contours are suitable for checking distant echoes. 

From the maps, profiles are prepared for various 
azimuths about the radar station. The first mile or so 
should be plotted accurately; at greater distances, 
the critical points, such as hills and breaks, should 
leceive the most attention. On each profile is drawn 
the tangent line from the center of the antenna to 
the point on the profile which determines the shield¬ 
ing, as in Figure 9. The angular elevation a of this 
line of sight is marked on the diagram. If a is nega¬ 
tive, the profile should be checked out to the radar 
horizon to obtain the correct shielding angle. On a 
plane earth diagram, the line of sight is actually 
curved, but for distances up to 10 miles it may be 
taken as straight with small error. The height 
difference, with a in radians, is then equal to 

h 2 — h = d tan a. (18) 

When the distance is larger so that earth curvature 
has to be taken into account, to the above expression 
for the height difference h 2 — hi must then be added 
the amount by which the earth is sloping av r ay over 
the distance d. This amount is d 2 /2ka, and the 
complete expression for h 2 — hi becomes 

h 2 — hi = d tan a + — . (19) 

2 ka 

For easier handling of this equation, a set of 
curves may be drawn where h 2 — hi is plotted against 
d for various constant values of a. These curves 
may then be used to determine the height of the 




















194 


SITING 


shielded region at any range. Thus all other moun¬ 
tains along the profile in Figure 9 might be checked 
for visibility by comparing the height of the moun¬ 
tain with the value of h 2 — hi read from the curve 
a = 0.5°. Any desired allowance for diffraction 
may be made by using a different curve such as 
a = 0. When the shield consists of several ridges 


lined. From an examination of the map, the azi¬ 
muths at which profiles should be prepared are 
determined. This will normally be about every 
10 degrees. Where the shielding is obviously good, 
the interval may be 20 degrees, but where the terrain 
is questionable, such as in a region of low hills, the 
profiles should be taken at 5-degree intervals. 



close together, an equivalent shield should be used. An overlay of the region is then prepared, showing 
This is derived by enclosing the ridges in a triangle, important geographical features and a polar-grid 
whose apex is taken as the shield (see Figure 6). system. On this chart is drawn the coverage contour 

The general procedure to be followed in preparing lines (broken lines in Figure 10). These lines repre- 
a prediction of permanent echoes will now be out- sent the limits of the heights of the shielded regions. 





























































EFFECTS OF TREES, JUNGLE, ETC. 


195 


Targets or mountains below and beyond the cover¬ 
age contours will not be visible except by diffraction. 
These contours may be drawn for several heights. 
Where they are close together, the shielding is good 
but the coverage is poor. Where the lines are widely 
separated, as toward the sea, there is little or no 
shielding except that due to earth curvature. With 
the coverage contour diagram superimposed on a 
map, the peaks exposed to radiation may be noted. 

The extent of the echoes due to these peaks de¬ 
pends, besides the size of the peak, on the horizontal 
radiation pattern, the pulse width, and the power and 
sensitivity of the radar. It should be noted that the 
half-power beamwidth is only a rough measure of the 
width of an echo and some greater angle between 
the half-power points and the nulls will usually be 
obtained for the echoes. 

The extension of the echo in range will be at least 
as great as the pulse width in miles as represented 
on the scope. This is about 0.1 mile per micro¬ 
second of pulse width. Actual echoes are thicker 
than this, since all the exposed hill sends back echoes. 

After a careful inspection of the profiles, taking 
into account the various factors mentioned above, the 
echoes are sketched in on the chart. In doing this, 
judgment and experience are important factors, but 
the following rules may be used as a guide. 

1. Shade in a circle for the main pulse several 
miles wide, depending on the pulse width and local 
return. 

2. Check each profile in turn and for each peak 
or hillside in front of the shielding ridge or mountain 
plot an echo for the main and all side lobes of the 
antenna. 

3. A series of sharp hills within the shielding part 
of the terrain should be plotted as a single large echo. 

4. The inner edge of an echo should be at the same 
range as the hill. 

5. Peaks beyond the shield may be in the diffrac¬ 
tion region and the relative strength of the echo 
may be estimated from a diffraction curve. 

6. In general, the echo strength varies as the 
inverse square of the distance and is roughly pro¬ 
portional to the target area. 

7. Where there is any doubt, the echo should be 
plotted. 

Experience is an essential factor in permanent 
echo prediction, regardless of the method used. The 
methods described here have been used successfully 
in many areas and are capable of accuracy adequate 
for most purposes. 


105 EFFECT OF TREES, JUNGLE, ETC. 

10 51 The Effect of Trees 

Trees form very effective obstacles for high-fre¬ 
quency radio weaves. A single tree may cause a drop 
in signal strength of several decibels. The attenua¬ 
tion is less for horizontal polarization than for 
vertical polarization for frequencies below 300 to 500 
megacycles. For higher frequencies, the polarization 
is not an important factor. With the transmitting 
antenna sited in a moderately wooded area, repre¬ 
sentative values for the losses are given in Table 1. 


Table 1 . Decrease in gain for transmitting antenna 
situated in a moderately wooded area. 



Horizontal 

Vertical 

Frequency 

polarization 

polarization 

30 me 

Negligible 

2- 3 db 

100 me 

1-2 db 

5-10 db 


When both antennas are in the woods these losses 
should be doubled. Measurements at 200 me for 
transmission through a grove of trees 100 feet wide 
show losses of 21 db for vertical polarization and 
6 db for horizontal polarization. 

When the antennas are in clearings, so that each is 
more than 200 or 300 feet from the edge of the woods, 
the decrease in gain is small. With vertical polariza¬ 
tion, there may be large and rapid variations of field 
intensity within a small area, due to reflections from 
near-by trees. 

10.5.2 The Effect of Jungles 

In jungles or heavy undergrowth, an exponential 
absorption is to be expected. Tests made of trans¬ 
mission through heavy jungles, such as are found in 
Panama or in New Guinea, show that the limit of 
transmission for ordinary field sets is 1 mile. An 
increase of power of several hundred fold is needed 
for a range of 2 miles. The decreases in gain en¬ 
countered are of the order of 50 to 60 db per mile. 

If the antennas are elevated above the jungle or 
located in clearings, the effect of the jungle may be 
minimized. Antennas should be 10 or more feet 
away from trees to avoid a change in antenna 
impedance. 

The best solution is sky-wave transmission even 
for distances as short as 1 mile. Due consideration 
should be given to the selection of optimum fre- 








19G 


SITING 


quencies based on ionosphere predictions. For 
distances up to 100 or 200 miles, a half-wave hori¬ 
zontal wire antenna should be used and the fre¬ 
quency range is about 2 to 8 me. The decrease in 
gain for the short path up through the trees is 
negligible at these frequencies. 


1° 5 3 Th e Effect of Trees and Obstacles 
on Microwaves 

At 10 cm, the absorption is so great with most 
objects that the diffracted energy is the principal 
portion transmitted. Only windows, light wooden 
walls, or branches of leafless trees show less than 


10 db loss. Opaque objects include: 

1. Rows of trees in leaf if more than two in depth. 

2. Screens of leafless trees if so dense that the 
skyline is invisible through them. 

3. Trunks of trees. 

4. Walls of masonry. 

5. Any but the lightest wooden buildings, espe¬ 
cially if there are partitions. 

Losses of a brick wall may be increased from 12 db 
to 46 db by wetting. In computing diffraction over 
treetops, the diffracting edge may be taken to be 
5 feet or so less in height. In a 1.25-cm test, the 
transmission loss through two medium-sized bare 
trees increased 18 db after leaves appeared. 



GLOSSARY 


a. 


A. 

Aq. 


Au 

Ap. 

A. 

b. 

B. 


1) Radius of the earth 

2) Radius of scattering plate, sphere, or cylinder 
Gain factor = A 0 Ap 

Gain factor for doublet antennas in free space, 
adjusted for maximum transfer of power = 
3\/8 t Td 

Plane-earth factor 
Path-gain factor 
Gain-factor curve parameter 

^ di — dj 
di -f- d 2 
Bandwidth 


c. 


C,C(v). 


1) * Velocity of light in free space 


hi + hi 

Real part of Fresnel’s integral 


d. 

do. 

do. 

di,d 2 . 

dr,dR. 

dL. 

dmax- 

D. 


e. 


E. 

E 0 . 


Ei. 


Distance from center of transmitting antenna to a 
point in space measured along the surface of the 
earth 

Free-space distance for field of strength E 


Normalized free-space distance = 

dT 

Distance from transmitter, receiver to reflecting 
point measured along the earth’s surface 
Distance from transmitter, receiver to the radio 
horizon measured along the earth’s surface 
Line of sight distance measured along the earth’s 
surface = dp + d# 

Maximum radar range 

1) Divergence factor for spherical earth 

2) Aperture of reflector 


1) Water-vapor pressure 

2) Coefficient for height-gain function 
Electric-field strength 

Maximum free-space field strength of a doublet 
transmitter at distance d 

Radiation field strength at one meter from trans¬ 
mitter 


Gr. 

h. 

hi,h 2 . 

hi',hi'. 

h c . 

ho. 

H. 

Hl. 

HA. 

I. 

Ii. 

3- 

k. 

K. 

I 

L. 


M. 


n',n" 

N. 

NF. 


Radar gain of a target 
Height above ground 

Height of transmitter, receiver above ground 
Height of transmitter, receiver above tangent plane 
at point of reflection 

Critical height distinguishing high and low antennas 
located in diffraction region = 30 X 2 / 3 
Virtual height of obstructing screen 

1) Magnetic field strength 

2) Height of a reflecting or diffracting obstruction 
Height-gain function for low antennas 

Hour angle of the sun 

RMS current 

Input current to antenna or circuit 

vri 

1) Boltzmann’s constant 

2) Factor multiplying earth’s radius to account for 

atmospheric refraction 

1) Amplitude of generalized reflection coefficient 

2) Echo constant of a target 

1) Length of a doublet 

2) Height coefficient to include effect of earth’s 

constants and wavelength 

1) Effective length of a doublet 

2) Characteristic length or scattering coefficient of 

a target 

3) Radar length of a target 

1) Ratio of radius of curvature of a ray to the radius 
of the earth = p a 


Aka{hi + h 2 ) 

Modified index of refraction 

1) Index of refraction 

2) Number of elements in an antenna array 
Lobe numbers 

Lobe variable for imperfect reflection 
Noise figure 


f. 1) Frequency 

2) Focal length of paraboloid reflector 
f c . Cutoff frequency of a wave guide 

f(h). Height-gain function 

f n . Height-gain function for the n th mode 

Fi,Fi. Fraction of maximum radiation field strength in the 
direction of direct, reflected rays 
F n . Noise figure 

F s . Shadow factor for the first mode 
F s r . Sum of shadow factors for all modes 

g. 1) Receiver gain 

2) Exponential factor of height-gain function for 
elevated antennas 
g'. Correction to g( 2) 

Gj. Transmitting antenna gain 

Gi. Receiving antenna gain 


p. 1) Total pressure of the atmosphere 
2) Dimensionless parameter = di/dT 

p' . Distance coefficient to include earth constants 

P. Power 

Pi. Power output of a transmitting doublet 
Pi. Power delivered by a receiving doublet to a matched 
load 

Pmin. Minimum power detectable by a receiver 
P n . Noise power 

P r . Power received by load circuit of receiving antenna 

P s . Scattered power 

q. Dimensionless parameter = d%/d 

Q. Parameter determining phase of beam reflected by 

the earth =—~ 






198 


GLOSSARY 


r. 

1) Distance from center of antenna to a point in 

space (usually replaced by d in applications) 

2) Height wavelength factor 

3) Pattern or chart parameter 

4) Path length of reflected ray 

8. 

1) Declination of the sun 

2) Angle of phase retardation due to path-length 

difference between direct and reflected rays 

3) Ground parameter depending on complex dielec¬ 

tric constant 

rd. 

Path length of direct ray 

A. 

Path-length difference between direct and reflected 

R. 

1) Resistance 


rays = r — r d 


2) Plane-earth reflection coefficient = 

A p . 

= ( 2 / 11 / 12 ) Id 


3) Path-difference parameter = ( kaA)/(hidT ) 

A(A P ). 

Correction factor for Ap 

Ra , 

Resistive component of antenna impedance 

A/. 

Bandwidth 

RH. 

Relative humidity in per cent 

V). 

Variable used in diffraction region 

R b 

Rr. 

Resistive component of load impedance 

Radiation resistance of an antenna 

€0- 

Dielectric constant of free space 

= 8.854 X HP 12 10- 9 

36x 

s. 

1) Spacing between dipoles in an antenna array 

2) Coefficient of distance for shadow factor 



3) Dimensionless coordinate = d\/d 

e C- 

Complex dielectric constant = e r — j e t - 

s. 

1) Scattering cross section 


Imaginary part of dielectric constant = 60 aX 


2) Area 

e r . 

Real part of dielectric constant 

S,S(v). 

Imaginary part of Fresnel’s integral 

f- 

1) Phase-angle lag due to diffraction 

t. 

1) Time 

2) Pulse width 


2) Dimensionless distance variable for curved-earth 
diffraction 


3) Degrees centigrade 

e. 

1) Angle between horizontal at transmitter base 

T. 

Absolute temperature 


and horizontal at point of reflection 

u. 

Dimensionless coordinate = b%/h\ 


2) Angle of tilt of scattering cylinder 

V. 

1) Velocity of wave propagation = c/n 

X. 

Wavelength 


2) Argument of Fresnel’s integral 

3) Dimensionless parameter = d/d? 

V. Voltage 

V n • Noise voltage 

W. Power per unit area 

Wi. Incident power per unit area 

W r . Scattered power per unit area at the receiver 

z. Amplitude of ratio of diffracted field to free-space 

field 

Z. Impedance 

Z a . Antenna impedance 

Zi. Load impedance 

a. 1) Attenuation constant, real part of propagation 

constant y 

2) Angle made by ray with the horizontal 
OL h a 2 . Angle of elevation of diffracting edge as seen from 
transmitter, receiver 
(3. 1) Bearing of the sun 

2) Phase constant, imaginary part of propagation 
constant y 

Pufii. Angle between ray from transmitter, receiver, and 
horizontal at diffracting edge 

y. 1) Propagation constant = <* + j/3 

2) Angle between horizontal at base of transmitter 
and line joining transmitter base to receiver 
y u Angle between the direct ray and the horizontal at 

the transmitter 


no. Permeability of free space = 4x10- 7 

nr- Permeability relative to free space 

v. Angle between reflected ray and horizontal at trans¬ 

mitter 

p. 1) Radius of curvature 

2) Amplitude of reflection coefficient 

a 1) Conductivity 

2) Radar cross section 

r. 1) Complex mode numbers 

2) Half beam-width angle 

</>. 1) Phase angle of reflection coefficient 

2) Latitude 

</)'. = </> — 7r = phase angle of reflection coefficient 

relative to a perfect reflector 
<!>,. Distance function for the first mode 

\p. 1) Phase-angle difference between currents in 

dipole-antenna array 

2) Angle between direct or reflected ray and the 

horizontal at the reflection point 

3) Height variable in the diffraction region 

\pd. Angle between direct ray and horizontal at reflec¬ 

tion point 

a?. Angular velocity = 2x/ 

12. Total phase lag between direct and reflected rays 

= < fi ' + 8 = ( f ) — 7T+5 




OSRD APPOINTEES 


COMMITTEE ON PROPAGATION 


Chairman 

Chas. R. Burrows 


H. H. Beverage 
T. J. Carroll 
J. H. Dellinger 


Members 

Martin Katzin 
D. E. Kerr 
J. A. Stratton 


Consultants 

S. S. Attwood J. A. Stratton 

C. E. Buell 


Technical Aides 

A. F. Murray S. W. Thomas 

R. J. Hearon 


CONTRACT NUMBERS, CONTRACTORS, AND SUBJECT OF CONTRACTS 


Contract No. 

Contractor 

Subject 

OEMsr-1207 

Columbia University- 

New York City, New York 

Correlation, analysis, and integration of data on radio and 
radar propagation. 

OEMsr-728 

State College of Washington 
Pullman, Washington 

Develop meteorological equipment and conduct meteor¬ 
ological soundings in the Southwest Pacific and correlate 
it with radio-propagation data. 

OEMsr-1497 

Humble Oil and Refining Co. 
Houston, Texas 

Development and construction of microwave field- 
strength measuring sets. 

OEMsr-1496 

University of Texas 

Austin, Texas 

Development of equipment for, and making measure¬ 
ments of, time and space deviations in radio-wave 
propagation. 

OEMsr-1502 

Jam Handy Organization, Inc. 
Detroit, Michigan 

Preparation of a General Outline of Training Material 
and the preparation of manuals, films, and other train¬ 
ing aids for use in instructing technical and other per¬ 
sonnel in radio-weather and radio propagation. 


200 






/ 


SERVICE PROJECTS 

The Committee on Propagation did all of its work under Project Control SOS-9, which was 
originally set up through the request of the Combined Chiefs of Staff following recommendations 
submitted by the Combined Meteorological Committee [CMC] (1): that the Committee on 
Propagation of the National Defense Research Committee be requested to act as a coordinating 
agency for all meteorological information associated with short-wave propagation, (2) that 
the Committee on Propagation be requested to forward periodically to the CMC a list of all 
reports and papers dealing with the meteorological aspects on short wave propagation which have 
been received or transmitted by that Committee. 

Later the Combined Meteorological Committee in its thirty-seventh meeting on Tuesday, 
February 22, 1944, agreed that the NDRC Committee on Propagation be recognized as the 
supervising committee on all basic research being done in the United States on the related prob¬ 
lems of radar propagation and weather, in addition it shall be the recognized channel whereby 
international exchange of papers of the two related sciences will be effected. 

The Joint Communications Board [JCB] therefore approved the following policy, 
which was concurred in by NDRC and by the Joint Meteorological Committee: 


1. The NDRC Propagation Committee and its associated working groups will initiate 
and exercise technical supervision over such tests and investigations as they deem necessary 
to ascertain the nature of the above mentioned propagation anomalies in the VHF, UHF, 
and SHF bands, to devise the most practicable methods to determine the occurrence and 
characteristics of these anomalies from appropriate meteorological forecasts, with a view to 
improving the interim solutions offered by the Joint Wave Propagation Committee of the 
JCB. 

2. The Army and Navy will furnish by direct coordination between them the basic staff 
guidance for such tests and investigations. They will accomplish this by determining (a) the 
specific forms in which basic prediction data shall be presented, and (6) the method of use 
required for operational forecast of propagation anomalies in the VHF, UHF, and SHF 
bands. 

3. When the NDRC requires the cooperation of the operating units of the Army and 
Navy in conducting such tests and investigations as it deems necessary and this cooperation 
is of such an extent and nature that it cannot be furnished by informal coordination, it will 
be requested through the Joint Wave Propagation Committee of the JCB. Such requests will 
be initiated by the NDRC representative on the Wave Propagation Committee and recom¬ 
mended to the JCB by the Joint Wave Propagation Committee for consideration. 

4. The Joint Wave Propagation Committee will be responsible for devising and furnish¬ 
ing immediately interim operational forecasting guides based upon information already 
available. 

On April 3, the Coordinator of Research and Development requested that the Army 
Project SOS-9 be made a joint Army-Navy project. Project No. AN-16 was assigned to this. 

On May 23, 1944, the Chief Signal Officer requested that under Project AN-16 the 
following work be inaugurated: 

Project AC 230.04, “Wave Propagation Study of Line-of-Sight Communication and 
Navigation.” 


201 




























INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. 
For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


A scope, calibration, 161-162 
Aircraft targets, radar cross section 
measurement, 185-186 
Antenna, general characteristics 
beam width, 22 

characteristics in transmission, 6 
diameter, 29-31 
effective length, 13 
function, 22 
horns, 44 

impedance of nearby conductors, 24 
pattern factors in ground reflection, 
23 

radiation patterns, 22-23 
radiation resistance, 24 
Antenna arrays, 33-39 
binomial, 38-39 
broadside, 34-38 
colinear, 34-38 
dipole, basic types, 34 
multidimensional, 38 
principle, 33-34 
ring, 39 

two-dipole side-by-side, 34-35 
unidirectional, 38 
Antenna gain 

calculation, 16-17 
definition, 16 
description, 22 

jungle locations, effect on gain, 195- 
196 

propagation factor in interference 
region, 69 

wood location, effect on gain, 195- 
196 

Antenna types, 22-33, 39-43 
directive antennas, 22-23 
multiple half-wave, 27-28 
parabolic reflector antennas, 42-43 
parasitic antennas, 39-42 
resonant antennas, 23 
standing-wave antennas, 23, 25-32 
traveling-wave antennas, 23-24, 32- 
33 

Atmospheric ducts 
effect on set performance, 163 
formation, 4 
types, 4 

Atmospheric stratification, effect on 
refraction, 50-52 
Attenuation, definition, 5 
Attenuation factors, radio gain calcu¬ 
lation, 61 

Beacons, performance characteristics, 
169 


Beam width, antennas, 22 
Binomial arrays, antenna, 38-39 
Brewster angle of reflection, defined, 54 
Broadside arrays, antenna, 34-38 
one-dimensional array, 35-38 
side-by-side array, 34 
unidirectional array, 38 

Calibration, A scope, 161-162 
Clarendon Laboratory at Oxford, con¬ 
ductivity of sea water, 55 
Coastal diffraction, transmission, 178- 
181 

cliff site, 181 
field strength, 179-181 
level site near coast, 178-179 
Colinear arrays, antenna, 34-38 
one-dimensional array, 38 
two half-wave dipole array, 35 
unidirectional array, 38 
Columbia University Wave Propaga¬ 
tion Group, 1 

Communication equipment, perform¬ 
ance characteristics, 169 
Conductivity, soil, 56-57 
Cophased dipole antenna, 28-29 
Corner-reflector antenna, 42 
Cornu’s spiral, diffraction theory, 172- 
173, 175 

Coverage diagrams, generalized coordi¬ 
nates, 144-159 
basic parameters, 144-145 
normalized free-space distance, 145- 
147 

path-difference parameter, 145, 147 
use of charts, 147-159 
Coverage diagrams, methods of con¬ 
struction, 132-144 
lobe-angle method, 138-144 
P-Q method, 132-135 
U-V method, 135-138 
Coverage diagrams, plane earth, 129- 
131 

field strength, 129 
horizontal polarization, 129-130 
vertical polarization, 130-131 
Coverage diagrams, spherical earth, 
131-132 

Curvature of earth 

calculation of radio gain below inter¬ 
ference region, 92 
geometrical relationships, 62 
Curvature of radio waves 

curvature relationships, 47-48 
diffraction by earth’s curvature, 58 
rays in standard atmosphere, 3-4 


Definition of propagation, 1 
Dielectric constant, soil, 56-57 
Dielectric constant, water, 54-55 
Dielectric earth 

graphical calculations, radio gain, 
95-108 

radiation field characteristics, 65-67 
sea water as dielectric earth, 66 
Diffraction 
by terrain, 170-181 
coastal diffraction, 178-181 
Cornu’s spiral, 172-173, 175 
definition, 58 

earth’s curvature diffraction, 58 
Fresnel diffraction theory, 170-175 
hill diffraction, 175-178 
raindrop diffraction, 59 
reflecting ground, 177-178 
slot diffraction, 175 
summary, 58 
target diffraction, 58-59 
transmission factor, 7 
Diffraction regions, radio gain calcula¬ 
tion, 62-63 

Dimensionless coordinates, radio gain 
calculations, 77-78 

Dimensionless parameters, radio gain 
calculation, 75-77 
Dipole arrays, basic types, 34 
Directive antennas, 22-23 
Divergence factor 
ground reflection, 57 
lobe length determination, 139-140 
propagation in interference region, 
68-69 

spherical earth radio gain calcula¬ 
tions, 75 

transmission characteristic, 7 
Doublet antennas, radio gain, 5 
Ducts, atmospheric 
effect on set performance, 163 
formation, 4 
types, 4 

Earth conductivity, 61 
Echo constant, radar coverage measure¬ 
ment, 182 

Effective length, antenna, 13 
Electric doublet antenna in free space, 
12-15 

radiation, 12-13 
received power, 13-14 
scattered power, 13-14 
transmission, 14-15 
End-fire arrays, antenna, 34 


203 


204 


INDEX 


Equipment performance, propagation 
aspects, 160-169 
A scope calibration, 161-162 
communications and radar equip¬ 
ment data, 169 

free space high-angle coverage, 162- 
163 

low-angle and surface coverage, 163- 
169 

performance figure, measurement of 
efficiency, 160 
reflection effect, 160-161 
signal-to-noise ratio, 161 
Equivalent height, radio gain calcula¬ 
tion, 71 

Field strength, definition, 5 
Free space distance, normalized, 145- 
147 

Free space field, definition, 5 
Free space gain factor 

below interference region, 91 
definition, 5 
equation, 60 

Free space radio gain, definition and 
formula, 15 

Fresnel diffraction theory, 170-175 
Fresnel integrals, 172-173 
Fresnel zones, 174-175 
mechanism of diffraction, 171 
polarization, 174 
slot diffraction, 175 
straight edge formula, 171-172 
Fresnel formulas, reflection, 54 
Fresnel-Kirchoff optical theory 
see Fresnel diffraction theory 

Generalized reflection coefficient, 69-70 
Grazing angle corresponding to lobe 
maxima, 129-130 

Grazing angle in radio gain calcula¬ 
tions, 78 

Ground reflection, 52-58 
analysis of problem, 52-53 
conductivity of soil, 56-57 
dielectric constant of soil, 56-57 
dielectric constant of water, 54-55 
divergence factor, 57 
Fresnel formulas, 54 
irregularity of ground, 57-58 
overland transmission, 56 
plane reflecting surface, 53 

Half-wave antennas, 25 
Half-wave dipole, antenna, 25-29 
comparison of alternate and cophased 
half-wave dipoles, 28 
cophased dipoles, 28-29 
folded dipole, 27 


gain, 26 
impedance, 26 

quarter-wave dipole, modification, 
26-27 

radiation field, 25-26 
radiation resistance, 26 
Hill diffraction, 175-178 

criterion, roughness of surface, 176 
field near the line of sight, 177 
reflecting ground diffraction, 177-178 
straight ridge diffraction, 176-177 
Horizontal polarization 

angles of lobe maxima, 129-130 
radio gain curves, 8-10 
Horns, antenna use, 44 
Huyghens principle, radio wave diffrac¬ 
tion, 7 

Impedance, half-wave dipole antenna, 
26 

Index of refraction, definition, 3 
Induction field, definition, 12 
Integral half-wave antennas, 27-28 
Interference, propagation factor in 
radio gain calculation, 67 
Isotropic radiator, hypothetical an¬ 
tenna, 16 

Jungles, obstacles to radio wave propa¬ 
gation, 195-196 

Line of sight, definition, 62 
Linear antennas, diameter lengths, 29- 
31 

Linear antennas, types, 24-25 
Lobe maxima and minima, definition, 
129 

Lobe-angle method, coverage diagram 
construction, 138-144 
basic equation, 138 
correction for low angles, 141-143 
lobe angles with horizontal, 139 
lobe construction, 140-141 
modified divergence factor, 139-140 
reflection point curves, 138-139 
vertical polarization, 143-144 

Maximum range, 162-166 

low-angle aircraft coverage, antenna 
heights and target heights, 163- 
164 

low-angle aircraft coverage, height 
curves versus maximum range, 
164 

low-angle and surface range coverage, 
163 

one-way communication, 162 
radar, 162 

Microwave beacons, ring arrays, 39 
Moist standard atmosphere, propaga¬ 
tion, 3-4 


Multidimensional arrays, antenna, 38 
Multiple half-wave long antennas, 27- 
28 

Noise figure of radio receiver, 17-19 
definition, 18 
measurement, 18-19 
Nonstandard atmosphere, propaga¬ 
tion, 4 

Operator loss, radar reception, 19 
Optical region, radiation field charac¬ 
teristics, 66 

Optical region, radio gain calculation, 
62-63 

Parabolic reflectors for antennas, 42-43 
Parasitic antennas, 39-42 
corner reflector, 42 
half-wave dipole and parasite, 39-41 
multiple parasites, Yagi antenna, 41 
reflecting screens, 41-42 
Path difference 

loci construction, coverage diagrams, 
134-135 

parameters, 145, 147 
path difference variable equation, 
80-81 

plane earth, 70 
spherical earth, 74-75 
Path gain factor, definition, 5 
Performance figure, measurement of set 
efficiency, 162-163 

Permanent echoes, prediction, 193-195 
profile method, 193-195 
radar test at site, 193 
RPD (radar planning device), 193 
supersonic method, 193 
Permanent echoes, site selection fac¬ 
tors, 191-195 

permanent echo diagrams, 191-192 
shielding, echo control, 193 
Plane earth 

calculation of radio gain below inter¬ 
ference region, 91-92 
coverage diagrams, 129-131 
path difference, 70 
Polarization 

angles of lobe maxima, horizontal 
polarization, 129-130 
angles of lobe maxima, vertical polar¬ 
ization, 130-131 
Fresnel diffraction theory, 174 
horizontal versus vertical, optical 
region, 66 

horizontal versus vertical, radio gain 
calculations, 109 

radio gain, vertical polarization 
effect, 83 

radio gain curves, horizontal polar¬ 
ization, 8-10 

radio wave diffraction, 174 



radio waves, 52-53 
transmission factor, 6 
Power transmission, 15-17 

antenna gain and polarization, 16-17 
radio gain, 15-16 
reciprocity principle, 17 
P-Q method of coverage diagram con¬ 
struction, 132-135 
path-difference loci construction, 
134-135 

range loci construction, 133-134 
Profile method, echo determination, 
193-195 

Propagation, assumption of standard 
conditions, 61-62 

Propagation, atmospheric considera¬ 
tions, 2-4 

moist standard atmosphere, 3-4 
nonstandard atmosphere, 4 
standard atmosphere, 3 
Propagation, basic relationships, 12-21 
electric doublet in free space, 12-15 
power transmission, 15-17 
radar cross section, 19-20 
radar gain, 20-21 
receiver sensitivity, 17-19 
Propagation, definition, 1 
Propagation, general characteristics, 
1-11 

atmospheric considerations, 2-4 
basic problems, 2 
radiation field characteristics, 7-8 
radio gain, 4-5, 8-10 
transmission factors, 6-7 
units and symbols used in propaga¬ 
tion study, 11 

Propagation aspects of low-angle and 
surface coverage performance, 

163- 169 

ducts, effect on set performance, 163 
low heights, effect on ranges, 163-164 
maximum range, 163 
maximum range versus height curves 

164- 166 

performance check before operation, 
166 

radar cross section of surface craft, 
164 

ship size estimation, 166 
Propagation below interference region, 
radio gain calculation, 91-128 
curved earth calculations, 92 
general problem analysis, 91-95 
graphical solution, dielectric earth 
calculations, 95-108 
plane earth calculations, 91-92 
radio gain near line of sight, 115 
sample calculations for general solu¬ 
tion, 116-128 

sea water calculations, 108-115 


INDEX 


Propagation in interference region, 
radio gain calculation, 66-91 
antenna gain and directivity, 69 
divergence, 68-69 
factors affecting radio gain, 67-69 
general solution, 69-70 
imperfect reflection, 67-68 
interference, 67 

plane earth calculations, 70-71 
sample calculations, 79-91 
spherical earth calculations, 71-78 
spreading effect, 67 

Quarter-wave dipole antenna, 26-27 

Radar cross section 
aircraft target, 185-186 
characteristic length L, 20 
maximum range calculations, 163 
scattering cross section, 19-20 
scattering parameters, radar cover¬ 
age measurement, 182 
surface craft, 164 

Radar cross section, simple forms, 183- 
185 

circular plate, 183-185 
corner reflector, 185 
cylinders, 183 
flat plates, 183 
rectangular plate, 185 
spheres, 183 
Radar gain, 5, 20-21 
Radar planning device for echo deter¬ 
mination (RPD), 193 
Radar receivers, sensitivity, 19 
operator loss, 19 
performance characteristics, 169 
scanning loss, 19 
sweep-speed loss, 19 
Radar siting 

see Siting, terrain selection and util¬ 
ization 

Radar targets, 182-186 

aircraft, cross section measurement, 
185-186 

radar cross section of simple forms, 
183-185 

scattering parameters, coverage 
measurement, 182-183 
Radiation 

antenna radiation patterns, 22-23 
antenna radiation resistance, 24 
electric doublet antenna, 12-13 
induction field, 12 

reciprocity principle with reception, 
17 

resistance, 13 

resistance, half-wave dipole, 26 
Radiation field 

electric doublet antenna, 12-13 
general nature, 7-8 
half-wave dipole antenna, 25-26 


205 


Radiation field in standard atmosphere, 
radio gain calculations, 63-67 
field variation, 63-65 
high antenna calculations, 65 
low antenna calculations, 65 
ultra short waves in diffraction re¬ 
gion, 65-67 
Radio gain 

basic equation, 15 
calculation, 60-128 
defined, 4-5, 60-61 
doublet antennas in free space, 5 
radio gain curves, 8-10 
Radio gain calculations, below inter¬ 
ference region, 91-128 
analysis of first mode, 91-94 
effect of linear variation of refractive 
index of atmosphere, 94-95 
general solution for dipole over a 
smooth sphere, 116-122 
graphical aids for sea water, v-h-f, 
vertical polarization, 108-115 
graphs for the case of the dielectric 
earth, 95-108 

radio gain near the line, 115 
sample calculation for very dry soil, 
123-128 

Radio gain calculations, general con¬ 
siderations, 60-63 
attenuation factors, 61 
curved-earth geometrical relation¬ 
ships, 62 
definition, 60-61 

optical and diffraction regions, 62-63 
standard propagation conditions as¬ 
sumed, 61-62 

Radio gain calculations, in interference 
region, 67-71 
general solution, 69-70 
plane earth, 70-71 
propagation factors, 67-69 
Radio gain calculations, in standard 
atmosphere, 63-67 

Radio gain calculations, optical-inter¬ 
ference region, 79-91 
coverage problem, 87-89 
for fixed heights and distance, 82-84 
maximum range vs. receiver height, 
89-91 

radio gain vs. distance for given an¬ 
tenna heights, 85-87 
radio gain vs. receiver height for 
given distance, 84-85 
Radio gain calculations, spherical 
earth, 71-78 
angle determination, 71 
dimensionless coordinates, 77-78 
dimensionless parameters, 75-77 
distance measurement, 71 
divergence factor, 75 
equivalent height, 71 





206 


INDEX 


grazing angle, 78 
path difference, 74-75 
reflection point determination, 71-74 
Raindrop diffraction, 59 
Range loci construction, 133-134 
Rayleigh criterion, radiation theory 
application, 176 
Receiver sensitivity, 17-19 
definition, 18 
noise figure, 17-19 
radar receivers, 19 
thermal noise, 17 

Reciprocity principle, reception and 
radiation, 17 
Reflection 

Brewster angle, 54 
effect on equipment performance, 
160-161 

ground reflection, 52-58 
imperfect reflection, interference re¬ 
gion, 67-68 

reflection coefficient, 53, 69-70 
transmission factors, 6-7 
Reflection point curves, coverage dia¬ 
gram construction, 138-139 
Reflection point determination, 71-74 
Reflection point variable, 80 
Refraction, 45-52 
atmospheric stratification, 50-52 
curvature relationships, 47-48 
definition, 3 

graphical representation, 46-47 
Snell’s law, 45-46 
standard refraction, 45 
transmission factor, 6 
Refractive index, 48-52 
computation, 48-50 
function of temperature and height, 
49-50 

function of temperature and relative 
humidity, 51 

modified refractive index, 46 
standard atmosphere, 61 
Resonant antennas, 23 
Rhombic antenna, traveling wave an¬ 
tenna, 32-33 
Ring arrays, antenna, 39 
RPD (radar planning device) for echo 
determination, 193 

Scanning loss, radar reception, 19 
Scattered power, radio wave reception, 
13-14 

Scattering parameters, radar coverage 
measurement, 182-183 


echo constant, 182 
equivalent plate area, 183 
radar cross section, 182 
scattering coefficient, radar length of 
target, 183 
target gain, 182 
Sea water 

dielectric earth behavior, 66-67 
influence on radio wave transmission, 
54-55 

radio gain calculation, 108-115 
Sectoral horn, antenna, 44 
Shadow zone, 7 

Shielding, permanent echo control de¬ 
vice, 193 

Ship roll, effect on radar coverage, 160- 
161 

Ship size estimated by strength of re¬ 
turned radar signal, 166 
Side-by-side array, two-dipole antenna, 
34-35 

Signal-to-noise ratio, radar equipment 
performance, 161 

Siting, terrain selection and utilization, 
187-196 

geometrical limits of visibility, 188- 
191 

permanent echoes, 191-195 
radar, overland transmission, 56 
requirements of siting, 187 
trees and jungles, 195-196 
Siting topography, 187-188 
maps, 187 

orientation, 187-188 
profiles, 187 
Slot diffraction, 175 
Snell’s law of refraction, 45-46 
Soil as radio wave conductor, 56-57 
Solar azimuths, calculation methods, 
187-188 
Spherical earth 

coverage diagrams, 131-132 
path difference, 74-75 
radio gain calculation; see Radio gain 
calculations, spherical earth 
Spreading effect, propagation factor in 
interference region, 67 
Standard atmosphere, refractive index, 
61 

Standard dry atmosphere, properties, 3 
Standard refraction, definition, 45 
Standing-wave antennas, 24-32 
half-wave dipole, 25-26, 28-29 
linear antennas, 24-25, 29-31 


multiple half-wave long antennas, 
27-28 

V antennas, 31-32 

Supersonic method of echo determina¬ 
tion, 193 

Surface craft, estimation of size by 
strength of returned radar sig¬ 
nal, 166 

Surface craft, radar cross section, 164 
Sweep-speed loss, radar reception, 19 
Symbols adopted for frequency ranges, 
11 

Target diffraction, 58-59 
Terrain diffraction, 170-181 
Terrain selection 

see Siting, terrain selection and util¬ 
ization 

Thermal noise in radio receivers, 17 
Time computations necessary for radar 
site selection, 188 
Topographic maps and profiles 
prediction of echoes, 193-195 
siting selection use, 187 
Transmission, 6-7, 45-59 
antenna characteristics, 6 
diffraction, 7, 58-59 
divergence, 7 
ground reflection, 52-58 
polarization, 6 
reflection, 6 
refraction, 6, 45-52 

Traveling-wave antennas, 23-24, 32-33 
Trees as obstacles for high frequency 
radio wave, 195-196 
Troposphere, propagation role, 2 

Ultra-short waves in diffraction region 
65-67 

Unidirectional arrays, antenna, 38 
Units used in propagation study, 11 
U-V method of coverage diagram con¬ 
struction, 135-138 

V antennas, 31-32 
Vertical polarization 

angles of lobe maxima, 130-131 
effect on radio gain, 83 
Visibility, geometrical limits, 188-191 
degree of shielding, 190-191 
horizon distance of transmitter, 
188-189 

obstacle height, 189-190 
Yagi antenna, multiple parasite, 41 










» 


41 



